### The Mizar article:

### Sequences in $\calE^N_\rmT$

**by****Agnieszka Sakowicz,****Jaroslaw Gryko, and****Adam Grabowski**

- Received May 10, 1994
Copyright (c) 1994 Association of Mizar Users

- MML identifier: TOPRNS_1
- [ MML identifier index ]

environ vocabulary FUNCT_1, PRE_TOPC, EUCLID, SEQ_1, RELAT_1, ANPROJ_1, BOOLE, ARYTM_1, COMPLEX1, FINSEQ_1, ABSVALUE, LATTICES, SEQ_2, ORDINAL2, ARYTM_3, NORMSP_1; notation TARSKI, XBOOLE_0, NUMBERS, XREAL_0, RELAT_1, FUNCT_1, REAL_1, NAT_1, SEQ_1, ABSVALUE, STRUCT_0, NORMSP_1, FINSEQ_1, FINSEQ_2, EUCLID, PRE_TOPC; constructors SEQ_1, ABSVALUE, EUCLID, REAL_1, NAT_1, PARTFUN1, NORMSP_1, MEMBERED, XBOOLE_0; clusters STRUCT_0, EUCLID, XREAL_0, RELSET_1, SEQ_1, ARYTM_3, MEMBERED, ZFMISC_1, XBOOLE_0, NUMBERS, ORDINAL2; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; theorems REAL_1, FUNCT_1, FUNCT_2, TARSKI, EUCLID, SEQ_1, NAT_1, AXIOMS, ABSVALUE, SEQ_2, XBOOLE_0, NORMSP_1, XCMPLX_0, XCMPLX_1; schemes SEQ_1, NAT_1, ZFREFLE1; begin definition let N be Nat; mode Real_Sequence of N is sequence of TOP-REAL N; end; reserve N,n,m,k,n1,n2 for Nat; reserve q,r,r1,r2 for Real; reserve x,y for set; reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N; reserve seq,seq1,seq2,seq3,seq' for Real_Sequence of N; canceled; theorem Th2: for f being Function holds f is Real_Sequence of N iff (dom f=NAT & for n holds f.n is Point of TOP-REAL N) proof let f be Function; thus f is Real_Sequence of N implies (dom f=NAT & for n holds f.n is Point of TOP-REAL N) by NORMSP_1:17; assume that A1: dom f=NAT and A2: for n holds f.n is Point of TOP-REAL N; for x holds x in NAT implies f.x is Point of TOP-REAL N by A2; hence thesis by A1,NORMSP_1:17; end; definition let N; let IT be Real_Sequence of N; attr IT is non-zero means :Def1: rng IT c= (the carrier of TOP-REAL N) \ {0.REAL N}; end; theorem Th3:seq is non-zero iff for x st x in NAT holds seq.x<>0.REAL N proof thus seq is non-zero implies for x st x in NAT holds seq.x<>0.REAL N proof assume seq is non-zero; then A1: rng seq c= (the carrier of TOP-REAL N) \ {0.REAL N} by Def1; let x; assume x in NAT; then x in dom seq by Th2; then seq.x in rng seq by FUNCT_1:def 5; then not seq.x in {0.REAL N} by A1,XBOOLE_0:def 4; hence seq.x<>0.REAL N by TARSKI:def 1; end; assume A2: for x st x in NAT holds seq.x<>0.REAL N; now let y; assume y in rng seq; then consider x such that A3: x in dom seq and A4: seq.x=y by FUNCT_1:def 5; A5: x in NAT by A3,NORMSP_1:17; then A6: y is Point of TOP-REAL N by A4,NORMSP_1:17; y<>0.REAL N by A2,A4,A5; then not y in {0.REAL N} by TARSKI:def 1; hence y in (the carrier of TOP-REAL N) \ {0.REAL N} by A6,XBOOLE_0:def 4 ; end; then rng seq c= (the carrier of TOP-REAL N) \ {0.REAL N} by TARSKI:def 3; hence thesis by Def1; end; theorem Th4:seq is non-zero iff for n holds seq.n<>0.REAL N proof thus seq is non-zero implies for n holds seq.n<>0.REAL N by Th3; assume for n holds seq.n<>0.REAL N; then for x holds x in NAT implies seq.x<>0.REAL N; hence thesis by Th3; end; theorem Th5:for N,seq,seq1 st (for x st x in NAT holds seq.x=seq1.x) holds seq=seq1 proof let N,seq,seq1 such that A1: for x st x in NAT holds seq.x=seq1.x; dom seq= NAT & dom seq1=NAT by NORMSP_1:17; hence thesis by A1,FUNCT_1:9; end; theorem Th6:for N,seq,seq1 st (for n holds seq.n=seq1.n) holds seq=seq1 proof let N,seq,seq1; assume for n holds seq.n=seq1.n; then for x holds x in NAT implies seq.x=seq1.x; hence thesis by Th5; end; scheme ExTopRealNSeq{N()->Nat, F(Nat)->Point of TOP-REAL N()}: ex seq being Real_Sequence of N() st for n holds seq.n=F(n) proof defpred P[set,set] means ex n st n=$1 & $2=F(n); A1: now let x; assume x in NAT; then consider n such that A2: n=x; reconsider r2=F(n) as set; take y=r2; thus P[x,y] by A2; end; consider f being Function such that A3: dom f=NAT and A4: for x st x in NAT holds P[x,f.x] from NonUniqFuncEx(A1); now let x; assume x in NAT; then ex n st n=x & f.x=F(n) by A4; hence f.x is Point of TOP-REAL N(); end; then reconsider f as Real_Sequence of N() by A3,NORMSP_1:17; take seq=f; let n; ex k st k=n & seq.n=F(k) by A4; hence seq.n=F(n); end; definition let N,seq1,seq2; func seq1 + seq2 -> Real_Sequence of N means :Def2: for n holds it.n = seq1.n + seq2.n; existence proof deffunc U(Nat) = seq1.$1 + seq2.$1; thus ex s being Real_Sequence of N st for n holds s.n = U(n) from ExTopRealNSeq; end; uniqueness proof let seq,seq' such that A1: for n holds seq.n=seq1.n+seq2.n and A2: for n holds seq'.n=seq1.n+seq2.n; now let n; seq.n=seq1.n+seq2.n & seq'.n=seq1.n+seq2.n by A1,A2; hence seq.n=seq'.n; end; hence seq=seq' by Th6; end; end; definition let r,N,seq; func r*seq -> Real_Sequence of N means : Def3: for n holds it.n = r*seq.n; existence proof deffunc U(Nat) = r * seq.$1; thus ex s being Real_Sequence of N st for n holds s.n = U(n) from ExTopRealNSeq; end; uniqueness proof let seq1,seq2 such that A1: for n holds seq1.n=r*seq.n and A2: for n holds seq2.n=r*seq.n; now let n; seq1.n=r*seq.n & seq2.n=r*seq.n by A1,A2; hence seq1.n=seq2.n; end; hence seq1=seq2 by Th6; end; end; definition let N,seq; func - seq -> Real_Sequence of N means :Def4: for n holds it.n = -seq.n; existence proof take seq1=(-1)*seq; let n; thus seq1.n=(-1)*seq.n by Def3 .=-(seq.n) by EUCLID:43; end; uniqueness proof let seq1,seq2 such that A1: for n holds seq1.n=-seq.n and A2: for n holds seq2.n=-seq.n; now let n; seq1.n=-seq.n & seq2.n=-seq.n by A1,A2; hence seq1.n=seq2.n; end; hence seq1=seq2 by Th6; end; end; definition let N,seq1,seq2; func seq1-seq2 -> Real_Sequence of N equals :Def5: seq1 +- seq2; correctness; end; definition let N; let x be Point of TOP-REAL N; func |.x.|-> Real means :Def6: ex y be FinSequence of REAL st x=y & it=|.y.|; existence proof reconsider y = x as FinSequence of REAL by EUCLID:27; take |.y.|; thus thesis; end; uniqueness; end; definition let N,seq; func |.seq.| -> Real_Sequence means :Def7: for n holds it.n=|.seq.n.|; existence proof deffunc U(Nat) = |.seq.$1.|; thus ex s being Real_Sequence st for n holds s.n= U(n) from ExRealSeq; end; uniqueness proof let seq8,seq9 be Real_Sequence such that A1: for n holds seq8.n=|.seq.n.| and A2: for n holds seq9.n=|.seq.n.|; now let n; seq9.n=|.seq.n.| by A2; hence seq8.n=seq9.n by A1; end; hence seq8=seq9 by FUNCT_2:113; end; end; canceled; theorem Th8: abs(r)*|.w.| = |.r*w.| proof reconsider s = w as Element of REAL N by EUCLID:25; r*w = r*s by EUCLID:def 11; then |.s.|=|.w.| & |.r*w.| = |.r*s.| by Def6; hence thesis by EUCLID:14; end; theorem |.r*seq.| = abs(r)(#)|.seq.| proof now let n; thus |.r*seq.|.n=|.(r*seq).n.| by Def7 .=|.r*(seq.n).| by Def3 .=abs(r)*|.seq.n.| by Th8 .=abs(r)*(|.seq.|).n by Def7 .=(abs(r)(#)|.seq.|).n by SEQ_1:13; end; hence thesis by FUNCT_2:113; end; theorem seq1 + seq2 = seq2 + seq1 proof now let n; thus (seq1+seq2).n = seq2.n + seq1.n by Def2 .= (seq2 + seq1).n by Def2; end; hence thesis by Th6; end; theorem Th11:(seq1 + seq2) + seq3 = seq1 + (seq2 + seq3) proof now let n; thus ((seq1+seq2)+seq3).n = (seq1+seq2).n+ seq3.n by Def2 .= seq1.n+seq2.n+seq3.n by Def2 .=seq1.n+(seq2.n+seq3.n) by EUCLID:30 .=seq1.n+(seq2+seq3).n by Def2 .=(seq1+(seq2+seq3)).n by Def2; end; hence thesis by Th6; end; theorem Th12:-seq = (-1)*seq proof now let n; thus ((-1)*seq).n=(-1)*seq.n by Def3 .=-seq.n by EUCLID:43 .=(-seq).n by Def4; end; hence thesis by Th6; end; theorem Th13: r*(seq1 + seq2) = r*seq1 + r*seq2 proof now let n; thus (r*(seq1 + seq2)).n=r*(seq1+seq2).n by Def3 .=r*(seq1.n+seq2.n) by Def2 .=r*seq1.n+r*seq2.n by EUCLID:36 .=(r*seq1).n+r*seq2.n by Def3 .=(r*seq1).n+(r*seq2).n by Def3 .=((r*seq1)+(r*seq2)).n by Def2; end; hence thesis by Th6; end; theorem Th14:(r*q)*seq = r*(q*seq) proof now let n; thus ((r*q)*seq).n=(r*q)*seq.n by Def3 .=r*(q*seq.n) by EUCLID:34 .=r*(q*seq).n by Def3 .=(r*(q*seq)).n by Def3; end; hence thesis by Th6; end; theorem Th15:r*(seq1 - seq2) = r*seq1 - r*seq2 proof thus r*(seq1-seq2)=r*(seq1+-seq2) by Def5 .=r*seq1+r*(-seq2) by Th13 .=r*seq1+r*((-1)*seq2) by Th12 .=r*seq1+((-1)*r)*seq2 by Th14 .=r*seq1+(-1)*(r*seq2) by Th14 .=r*seq1+-(r*seq2) by Th12 .=r*seq1-(r*seq2) by Def5; end; theorem seq1 - (seq2 + seq3) = seq1 - seq2 - seq3 proof thus seq1-(seq2+seq3)=seq1+-(seq2+seq3) by Def5 .=seq1+(-1)*(seq2+seq3) by Th12 .=seq1+((-1)*seq2+(-1)*seq3) by Th13 .=seq1+(-seq2+(-1)*seq3) by Th12 .=seq1+(-seq2+-seq3) by Th12 .=seq1+-seq2+-seq3 by Th11 .=seq1-seq2+-seq3 by Def5 .=seq1-seq2-seq3 by Def5; end; theorem Th17:1*seq=seq proof now let n; thus (1*seq).n=1*seq.n by Def3 .=seq.n by EUCLID:33; end; hence thesis by Th6; end; theorem Th18:-(-seq) = seq proof thus -(-seq)=(-1)*(-seq) by Th12 .=(-1)*((-1)*seq) by Th12 .=((-1)*(-1))*seq by Th14 .=seq by Th17; end; theorem Th19:seq1 - (-seq2) = seq1 + seq2 proof thus seq1-(-seq2)=seq1+(-(-seq2)) by Def5 .=seq1+seq2 by Th18; end; theorem seq1 - (seq2 - seq3) = seq1 - seq2 + seq3 proof thus seq1-(seq2-seq3)=seq1+-(seq2-seq3) by Def5 .=seq1+(-1)*(seq2-seq3) by Th12 .=seq1+((-1)*seq2-((-1)*seq3)) by Th15 .=seq1+(-seq2-((-1)*seq3)) by Th12 .=seq1+(-seq2-(-seq3)) by Th12 .=seq1+(-seq2+seq3) by Th19 .=seq1+-seq2+seq3 by Th11 .=seq1-seq2+seq3 by Def5; end; theorem seq1 + (seq2 - seq3) = seq1 + seq2 - seq3 proof thus seq1+(seq2-seq3)=seq1+(seq2+-seq3) by Def5 .=seq1+seq2+-seq3 by Th11 .=seq1+seq2-seq3 by Def5; end; theorem Th22:r <> 0 & seq is non-zero implies r*seq is non-zero proof assume that A1: r<>0 and A2: seq is non-zero and A3: not r*seq is non-zero; consider n1 such that A4: (r*seq).n1=0.REAL N by A3,Th4; A5: r*seq.n1=0.REAL N by A4,Def3; seq.n1 <> 0.REAL N by A2,Th4; hence contradiction by A1,A5,EUCLID:35; end; theorem seq is non-zero implies -seq is non-zero proof assume seq is non-zero; then (-1)*seq is non-zero by Th22; hence thesis by Th12; end; theorem Th24:|.0.REAL N.| = 0 proof 0.REAL N = 0*N by EUCLID:def 9; hence |.0.REAL N.| = |.0*N.| by Def6 .=0 by EUCLID:10; end; theorem Th25:|.w.| = 0 implies w = 0.REAL N proof reconsider s = w as Element of REAL N by EUCLID:25; assume |.w.| = 0; then |.s.| = 0 by Def6; then s = 0*N by EUCLID:11 .= 0.REAL N by EUCLID:def 9; hence thesis; end; theorem Th26:|.w.| >= 0 proof reconsider s = w as Element of REAL N by EUCLID:25; |.s.| >= 0 by EUCLID:12; hence thesis by Def6; end; theorem Th27:|.-w.| = |.w.| proof reconsider s = w as Element of REAL N by EUCLID:25; -s = -w by EUCLID:def 12; then |.-w.| = |.-s.| by Def6 .=|.s.| by EUCLID:13 .=|.w.| by Def6; hence thesis; end; theorem Th28:|.w1 - w2.| = |.w2 - w1.| proof thus |.w1 - w2.| = |.-(w1 - w2).| by Th27 .= |.w2 - w1.| by EUCLID:48; end; Lm1: |.w1 - w2.| = 0 implies w1 = w2 proof assume |.w1 - w2.| = 0; then w1 - w2 = 0.REAL N by Th25; hence w1 = w2 by EUCLID:47; end; Lm2: w1 = w2 implies |.w1 - w2.| = 0 proof assume w1 = w2; then |.w1 - w2.| = |.0.REAL N.| by EUCLID:46 .= 0 by Th24; hence thesis; end; theorem |.w1 - w2.| = 0 iff w1 = w2 by Lm1,Lm2; theorem Th30:|.w1 + w2.| <= |.w1.| + |.w2.| proof reconsider s1 = w1, s2 = w2 as Element of REAL N by EUCLID:25; w1 + w2 = s1 + s2 by EUCLID:def 10; then A1: |.w1 + w2.| = |.s1 + s2.| by Def6; A2: |.s1 + s2.| <= |.s1.| + |.s2.| by EUCLID:15; |.s1.| = |.w1.| by Def6; hence |.w1 + w2.| <= |.w1.| + |.w2.| by A1,A2,Def6; end; theorem |.w1 - w2.| <= |.w1.| + |.w2.| proof reconsider s1 = w1, s2 = w2 as Element of REAL N by EUCLID:25; s1 - s2 = w1 - w2 by EUCLID:def 13; then A1: |.w1 - w2.| = |.s1 - s2.| by Def6; A2: |.s1 - s2.| <= |.s1.| + |.s2.| by EUCLID:16; |.s1.| = |.w1.| by Def6; hence |.w1 - w2.| <= |.w1.| + |.w2.| by A1,A2,Def6; end; theorem |.w1.| - |.w2.| <= |.w1 + w2.| proof reconsider s1 = w1, s2 = w2 as Element of REAL N by EUCLID:25; s1 + s2 = w1 + w2 by EUCLID:def 10; then A1: |.w1 + w2.| = |.s1 + s2.| by Def6; A2: |.s1.| - |.s2.| <= |.s1 + s2.| by EUCLID:17; |.s1.| = |.w1.| by Def6; hence |.w1.| - |.w2.| <= |.w1 + w2.| by A1,A2,Def6; end; theorem Th33:|.w1.| - |.w2.| <= |.w1 - w2.| proof reconsider s1 = w1, s2 = w2 as Element of REAL N by EUCLID:25; s1 - s2 = w1 - w2 by EUCLID:def 13; then A1: |.w1 - w2.| = |.s1 - s2.| by Def6; A2: |.s1.| - |.s2.| <= |.s1 - s2.| by EUCLID:18; |.s1.| = |.w1.| by Def6; hence |.w1.| - |.w2.| <= |.w1 - w2.| by A1,A2,Def6; end; theorem w1 <> w2 implies |.w1 - w2.| > 0 proof reconsider s1 = w1, s2 = w2 as Element of REAL N by EUCLID:25; A1: s1 - s2 = w1 - w2 by EUCLID:def 13; assume w1 <> w2; then |.s1 - s2.| > 0 by EUCLID:20; hence thesis by A1,Def6; end; theorem |.w1 - w2.| <= |.w1 - w.| + |.w - w2.| proof 0.REAL N = w - w by EUCLID:46 .= -w + w by EUCLID:45; then |.w1 - w2.| = |.w1 + ((-w) + w) - w2.| by EUCLID:31 .= |.w1 + (-w) + w - w2.| by EUCLID:30 .= |.(w1 - w) + w - w2.| by EUCLID:45 .= |.(w1 - w) + (w - w2).| by EUCLID:49; hence thesis by Th30; end; theorem 0<=r1 & |.w1.|<|.w2.| & r1<r2 implies |.w1.|*r1<|.w2.|*r2 proof assume that A1: 0 <=r1 and A2: |.w1.|<|.w2.| and A3: r1<r2; A4: |.w1.|*r2<|.w2.|*r2 by A1,A2,A3,REAL_1:70; 0 <= |.w1.| by Th26; then |.w1.|*r1<=|.w1.|*r2 by A3,AXIOMS:25; hence thesis by A4,AXIOMS:22; end; canceled; theorem -|.w.|<r & r<|.w.| iff abs(r)<|.w.| proof thus -|.w.|<r & r<|.w.| implies abs(r)<|.w.| proof assume that A1: -|.w.|<r and A2: r<|.w.|; A3: abs(r)<=|.w.| by A1,A2,ABSVALUE:12; now assume A4: abs(r)=|.w.|; now assume r<0; then -|.w.|=--r by A4,ABSVALUE:def 1; hence contradiction by A1; end; hence contradiction by A2,A4,ABSVALUE:def 1; end; hence thesis by A3,REAL_1:def 5; end; assume A5: abs(r)<|.w.|; then A6: -|.w.|<=r by ABSVALUE:12; A7: r<=|.w.| by A5,ABSVALUE:12; A8: 0<=abs(r) by ABSVALUE:5; A9: 0<|.w.| by A5,ABSVALUE:5; A10: -|.w.|<0 by A5,A8,REAL_1:26,50; now assume r=-|.w.|; then abs(r)=--|.w.| by A10,ABSVALUE:def 1 .=|.w.|; hence contradiction by A5; end; hence -|.w.|<r by A6,REAL_1:def 5; r <> |.w.| by A5,A9,ABSVALUE:def 1; hence r<|.w.| by A7,REAL_1:def 5; end; definition let N; let IT be Real_Sequence of N; attr IT is bounded means :Def8: ex r st for n holds |.IT.n.| < r; end; theorem Th39:for n ex r st (0<r & for m st m<=n holds |.seq.m.| < r) proof defpred X[Nat] means ex r1 st 0<r1 & for m st m<=$1 holds |.seq.m.|<r1; A1: X[0] proof take r=|.seq.0.|+1; 0<=|.seq.0.| by Th26; then 0+0<|.seq.0.|+1 by REAL_1:67; hence 0<r; let m such that A2: m<=0; 0=m by A2,NAT_1:18; then |.seq.m.|+0<r by REAL_1:67; hence |.seq.m.|<r; end; A3: for n st X[n] holds X[n+1] proof let n; given r1 such that A4: 0<r1 and A5: for m st m<=n holds |.seq.m.|<r1; A6: now assume A7: |.seq.(n+1).|<=r1; take r=r1+1; 0+0<r1+1 by A4,REAL_1:67; hence 0<r; let m such that A8: m<=n+1; A9: now assume m<=n; then A10: |.seq.m.|<r1 by A5; r1+0<r by REAL_1:67; hence |.seq.m.|<r by A10,AXIOMS:22; end; now assume A11: m=n+1; r1+0<r by REAL_1:67; hence |.seq.m.|<r by A7,A11,AXIOMS:22; end; hence |.seq.m.|<r by A8,A9,NAT_1:26; end; now assume A12: r1<=|.seq.(n+1).|; take r=|.seq.(n+1).|+1; 0<=|.seq.(n+1).| by Th26; then 0+0<r by REAL_1:67; hence 0<r; let m such that A13: m<=n+1; A14: now assume m<=n; then |.seq.m.|<r1 by A5; then |.seq.m.|<|.seq.(n+1).| by A12,AXIOMS:22; then |.seq.m.|+0<r by REAL_1:67; hence |.seq.m.|<r; end; now assume m=n+1; then |.seq.m.|+0<r by REAL_1:67; hence |.seq.m.|<r; end; hence |.seq.m.|<r by A13,A14,NAT_1:26; end; hence ex r st (0<r & for m st m<=n+1 holds |.seq.m.|<r) by A6; end; thus for n holds X[n] from Ind(A1,A3); end; definition let N; let IT be Real_Sequence of N; attr IT is convergent means :Def9: ex g st for r st 0<r ex n st for m st n<=m holds |.IT.m-g.| < r; end; definition let N,seq; assume A1: seq is convergent; func lim seq -> Point of TOP-REAL N means :Def10: for r st 0<r ex n st for m st n<=m holds |.seq.m-it.| < r; existence by A1,Def9; uniqueness proof given g1,g2 such that A2: for r st 0<r ex n st for m st n<=m holds |.seq.m-g1.|<r and A3: for r st 0<r ex n st for m st n<=m holds |.seq.m-g2.|<r and A4: g1<>g2; A5: now assume |.g1-g2.|=0; then 0.REAL N+g2=g1-g2+g2 by Th25; then g2=g1-g2+g2 by EUCLID:31 .=g1-(g2-g2) by EUCLID:51 .=g1-0.REAL N by EUCLID:46 .=g1+ -0.REAL N by EUCLID:45 .=g1+(-1)*0.REAL N by EUCLID:43 .=g1+0.REAL N by EUCLID:32 .=g1 by EUCLID:31; hence contradiction by A4; end; A6: 0<=|.g1-g2.| by Th26; then A7: 0<|.g1-g2.|/4 by A5,SEQ_2:3; then consider n1 such that A8: for m st n1<=m holds |.seq.m-g1.|<|.g1-g2.|/4 by A2; consider n2 such that A9: for m st n2<=m holds |.seq.m-g2.|<|.g1-g2.|/4 by A3,A7; n1<=n1+n2 by NAT_1:37; then A10: |.seq.(n1+n2)-g1.|<|.g1-g2.|/4 by A8; n2<=n1+n2 by NAT_1:37; then |.seq.(n1+n2)-g2.|<|.g1-g2.|/4 by A9; then |.seq.(n1+n2)-g2.|+|.seq.(n1+n2)-g1.|<|.g1-g2.|/4+|.g1-g2.|/4 by A10,REAL_1:67; then A11: |.seq.(n1+n2)-g1.|+|.seq.(n1+n2)-g2.|<|.g1-g2.|/2 by XCMPLX_1:72; |.g1-g2.|=|.g1-g2+0.REAL N.| by EUCLID:31 .=|.g1-g2+(-1)*0.REAL N.| by EUCLID:32 .= |.g1-g2+-0.REAL N.| by EUCLID:43 .= |.g1-g2-0.REAL N.| by EUCLID:45 .=|.g1-g2-(seq.(n1+n2)-seq.(n1+n2)).| by EUCLID:46 .=|.g1-g2-seq.(n1+n2)+seq.(n1+n2).| by EUCLID:51 .=|.g1-(seq.(n1+n2)+g2)+seq.(n1+n2).| by EUCLID:50 .=|.g1-seq.(n1+n2)-g2+seq.(n1+n2).| by EUCLID:50 .=|.g1-seq.(n1+n2)-(g2-seq.(n1+n2)).| by EUCLID:51 .=|.g1-seq.(n1+n2)+-(g2-seq.(n1+n2)).| by EUCLID:45 .=|.g1-seq.(n1+n2)+(seq.(n1+n2)-g2).| by EUCLID:48 .=|.-(seq.(n1+n2)-g1)+(seq.(n1+n2)-g2).| by EUCLID:48; then |.g1-g2.|<=|.-(seq.(n1+n2)-g1).|+|.seq.(n1+n2)-g2.| by Th30; then A12: |.g1-g2.|<=|.seq.(n1+n2)-g1.|+|.seq.(n1+n2)-g2.| by Th27; |.g1-g2.|/2 <|.g1-g2.| by A5,A6,SEQ_2:4; hence contradiction by A11,A12,AXIOMS:22; end; end; canceled; theorem Th41:seq is convergent & seq' is convergent implies seq + seq' is convergent proof assume that A1: seq is convergent and A2: seq' is convergent; consider g1 such that A3: for r st 0<r ex n st for m st n<=m holds |.seq.m-g1.|<r by A1,Def9; consider g2 such that A4: for r st 0<r ex n st for m st n<=m holds |.seq'.m-g2.|<r by A2,Def9; take g=g1+g2; let r;assume 0<r; then A5: 0<r/2 by SEQ_2:3; then consider n1 such that A6: for m st n1<=m holds |.seq.m-g1.|<r/2 by A3; consider n2 such that A7: for m st n2<=m holds |.seq'.m-g2.|<r/2 by A4,A5; take k=n1+n2; let m; assume A8: k<=m; n1<=n1+n2 by NAT_1:37; then n1<=m by A8,AXIOMS:22; then A9: |.seq.m-g1.|<r/2 by A6; n2<=k by NAT_1:37; then n2<=m by A8,AXIOMS:22; then |.seq'.m-g2.|<r/2 by A7; then |.seq.m-g1.|+|.seq'.m-g2.|<r/2+r/2 by A9,REAL_1:67; then A10: |.seq.m-g1.|+|.seq'.m-g2.|<r by XCMPLX_1:66; A11:|.(seq+seq').m-g.|=|.seq.m+seq'.m-(g1+g2).| by Def2 .=|.seq.m+(seq'.m-(g1+g2)).| by EUCLID:49 .=|.seq.m+((-(g1+g2))+seq'.m).| by EUCLID:45 .=|.seq.m+-(g1+g2)+seq'.m.| by EUCLID:30 .=|.seq.m-(g1+g2)+seq'.m.| by EUCLID:45 .=|.seq.m-g1-g2+seq'.m.| by EUCLID:50 .=|.seq.m-g1+-g2+seq'.m.| by EUCLID:45 .=|.seq.m-g1+(seq'.m+-g2).| by EUCLID:30 .=|.seq.m-g1+(seq'.m-g2).| by EUCLID:45; |.seq.m-g1+(seq'.m-g2).|<=|.seq.m-g1.|+|.seq'.m-g2.| by Th30; hence |.(seq+seq').m -g.|<r by A10,A11,AXIOMS:22; end; theorem Th42:seq is convergent & seq' is convergent implies lim (seq + seq')=(lim seq)+(lim seq') proof assume that A1: seq is convergent and A2: seq' is convergent; A3: seq+seq' is convergent by A1,A2,Th41; now let r; assume 0<r; then A4: 0<r/2 by SEQ_2:3; then consider n1 such that A5: for m st n1<=m holds |.seq.m-lim (seq).|<r/2 by A1,Def10; consider n2 such that A6: for m st n2<=m holds |.seq'.m-lim (seq').|<r/2 by A2,A4,Def10; take k=n1+n2; let m such that A7: k<=m; n1<=n1+n2 by NAT_1:37; then n1<=m by A7,AXIOMS:22; then A8: |.seq.m-lim(seq).|<r/2 by A5; n2<=k by NAT_1:37; then n2<=m by A7,AXIOMS:22; then |.seq'.m-(lim seq').|<r/2 by A6; then |.seq.m-(lim seq).|+|.seq'.m-(lim seq').|<r/2+r/2 by A8,REAL_1:67; then A9: |.seq.m-(lim seq).|+|.seq'.m-(lim seq').|<r by XCMPLX_1:66; A10: |.((seq+seq').m)-((lim seq)+(lim seq')).| =|.((seq+seq').m)-(lim seq)-(lim seq').| by EUCLID:50 .=|.seq.m+seq'.m-(lim seq)-(lim seq').| by Def2 .=|.seq.m+(seq'.m-(lim seq))-(lim seq').| by EUCLID:49 .=|.seq.m+(-(lim seq)+seq'.m)-(lim seq').| by EUCLID:45 .=|.seq.m+-(lim seq)+seq'.m-(lim seq').| by EUCLID:30 .=|.seq.m-(lim seq)+seq'.m-(lim seq').| by EUCLID:45 .=|.seq.m-(lim seq)+(seq'.m-(lim seq')).| by EUCLID:49; |.seq.m-(lim seq)+(seq'.m-(lim seq')).|<= |.seq.m-(lim seq).|+|.seq'.m-(lim seq').| by Th30; hence |.((seq+seq').m)-((lim seq)+(lim seq')).|<r by A9,A10,AXIOMS:22; end; hence thesis by A3,Def10; end; theorem Th43:seq is convergent implies r*seq is convergent proof assume seq is convergent; then consider g1 such that A1: for q st 0<q ex n st for m st n<=m holds |.seq.m-g1.|<q by Def9; set g=r*g1; A2: now assume A3: r=0; let q such that A4: 0<q; take k=0; let m such that k<=m; |.((r*seq).m)-g.|=|.((0 *seq).m)-0.REAL N.| by A3,EUCLID:33 .=|.0.REAL N-((0 * seq).m).| by Th28 .=|.0.REAL N+-((0 * seq).m).| by EUCLID:45 .=|.-((0 * seq).m).| by EUCLID:31 .=|.(0 * seq).m.| by Th27 .=|.0 * seq.m.| by Def3 .=|.0.REAL N.| by EUCLID:33 .=0 by Th24; hence |.((r*seq).m)-g.|<q by A4; end; now assume A5: r<>0; then A6: 0<abs(r) by ABSVALUE:6; let q such that A7:0<q; A8: 0<abs(r) by A5,ABSVALUE:6; 0/abs(r)=0; then 0<q/abs(r) by A6,A7,REAL_1:73; then consider n1 such that A9: for m st n1<=m holds |.seq.m-g1.|<q/abs(r) by A1; take k=n1; let m; assume k<=m; then A10: |.seq.m-g1.|<q/abs(r) by A9; A11: |.((r*seq).m)-g.|=|.r*seq.m-r*g1.| by Def3 .=|.r*(seq.m-g1).| by EUCLID:53 .=abs(r)*|.seq.m-g1.| by Th8; abs(r)*(q/abs(r))=abs(r)*(abs(r)"*q) by XCMPLX_0:def 9 .=abs(r)*abs(r)"*q by XCMPLX_1:4 .=1*q by A8,XCMPLX_0:def 7 .=q; hence |.((r*seq).m)-g.|<q by A6,A10,A11,REAL_1:70; end; hence thesis by A2,Def9; end; theorem Th44:seq is convergent implies lim(r*seq)=r*(lim seq) proof assume A1: seq is convergent; then A2: r*seq is convergent by Th43; A3: now assume A4: r=0; then A5: r*(lim seq)=0.REAL N by EUCLID:33; let q such that A6: 0<q; take k=0; let m such that k<=m; |.((r*seq).m)-r*(lim seq).|=|.0 * seq.m-0.REAL N.| by A4,A5,Def3 .=|.0.REAL N-0 * seq.m.| by Th28 .=|.0.REAL N+-0 * seq.m.| by EUCLID:45 .=|.-0 * seq.m.| by EUCLID:31 .=|.0 * seq.m.| by Th27 .=|.0.REAL N.| by EUCLID:33 .=0 by Th24; hence |.((r*seq).m)-r*(lim seq).|<q by A6; end; now assume A7: r<>0; then A8: 0<abs(r) by ABSVALUE:6; let q such that A9: 0<q; A10: 0<>abs(r) by A7,ABSVALUE:6; 0/abs(r)=0; then 0<q/abs(r) by A8,A9,REAL_1:73; then consider n1 such that A11: for m st n1<=m holds |.seq.m-(lim seq).|<q/abs(r) by A1,Def10; take k=n1; let m; assume k<=m; then A12: |.seq.m-(lim seq).|<q/abs(r) by A11; A13: |.((r*seq).m)-r*(lim seq).|=|.r*seq.m-r*(lim seq).| by Def3 .=|.r*(seq.m-(lim seq)).| by EUCLID:53 .=abs(r)*|.seq.m-(lim seq).| by Th8; abs(r)*(q/abs(r))=abs(r)*(abs(r)"*q) by XCMPLX_0:def 9 .=abs(r)*abs(r)"*q by XCMPLX_1:4 .=1*q by A10,XCMPLX_0:def 7 .=q; hence |.((r*seq).m)-r*(lim seq).|<q by A8,A12,A13,REAL_1:70; end; hence thesis by A2,A3,Def10; end; theorem Th45:seq is convergent implies - seq is convergent proof assume seq is convergent; then (-1)*seq is convergent by Th43; hence thesis by Th12; end; theorem Th46:seq is convergent implies lim(-seq)=-(lim seq) proof assume seq is convergent; then lim ((-1)*seq)=(-1)*(lim seq) by Th44 .=-(1*(lim seq)) by EUCLID:44 .=-(lim seq) by EUCLID:33; hence thesis by Th12; end; theorem seq is convergent & seq' is convergent implies seq - seq' is convergent proof assume that A1: seq is convergent and A2: seq' is convergent; -seq' is convergent by A2,Th45; then seq+-seq' is convergent by A1,Th41; hence thesis by Def5; end; theorem seq is convergent & seq' is convergent implies lim(seq - seq')=(lim seq)-(lim seq') proof assume that A1: seq is convergent and A2: seq' is convergent; A3: - seq' is convergent by A2,Th45; thus lim(seq - seq')=lim (seq +(-seq')) by Def5 .=(lim seq)+(lim(- seq')) by A1,A3,Th42 .=(lim seq)+-(lim seq') by A2,Th46 .=(lim seq)-(lim seq') by EUCLID:45; end; canceled; theorem seq is convergent implies seq is bounded proof assume seq is convergent; then consider g such that A1: for r st 0<r ex n st for m st n<=m holds |.seq.m-g.|<r by Def9; consider n1 such that A2: for m st n1<=m holds |.seq.m-g.|<1 by A1; A3: now take r=|.g.|+1; 0<=|.g.| by Th26; then 0+0<r by REAL_1:67; hence 0<r; let m; assume n1<=m; then A4: |.seq.m-g.|<1 by A2; |.seq.m.|-|.g.|<=|.seq.m-g.| by Th33; then A5: |.seq.m.|-|.g.|<1 by A4,AXIOMS:22; |.seq.m.|-|.g.|+|.g.|=|.seq.m.|-(|.g.|-|.g.|) by XCMPLX_1:37 .=|.seq.m.|-(|.g.|+-|.g.|) by XCMPLX_0:def 8 .=|.seq.m.|-0 by XCMPLX_0:def 6 .=|.seq.m.|; hence |.seq.m.|<r by A5,REAL_1:53; end; now consider r1 such that A6: 0<r1 and A7: for m st n1<=m holds |.seq.m.|<r1 by A3; consider r2 such that A8: 0<r2 and A9: for m st m<=n1 holds |.seq.m.|<r2 by Th39; take r=r1+r2; 0+0<r by A6,A8,REAL_1:67; hence 0<r; A10: r1+0<r by A8,REAL_1:67; A11: 0+r2<r by A6,REAL_1:67; let m; A12: now assume n1<=m; then |.seq.m.|<r1 by A7; hence |.seq.m.|<r by A10,AXIOMS:22; end; now assume m<=n1; then |.seq.m.|<r2 by A9; hence |.seq.m.|<r by A11,AXIOMS:22; end; hence |.seq.m.|<r by A12; end; hence thesis by Def8; end; theorem seq is convergent implies ((lim seq) <> 0.REAL N implies ex n st for m st n<=m holds |.(lim seq).|/2 < |.seq.m.|) proof assume that A1: seq is convergent and A2: (lim seq)<>0.REAL N; A3: 0<=|.(lim seq).| by Th26; 0 <> |.(lim seq).| by A2,Th25; then 0<|.(lim seq).|/2 by A3,SEQ_2:3; then consider n1 such that A4: for m st n1<=m holds |.seq.m-(lim seq).|<|.(lim seq).|/2 by A1,Def10; take n=n1; let m; assume n<=m; then A5: |.seq.m-(lim seq).|<|.(lim seq).|/2 by A4; A6: |.(lim seq).|-|.seq.m.|<=|.(lim seq)-seq.m.| by Th33; |.(lim seq)-seq.m.|=|.seq.m-(lim seq).| by Th28; then A7: |.(lim seq).|-|.seq.m.|<|.(lim seq).|/2 by A5,A6,AXIOMS:22; A8: |.(lim seq).|/2+(|.seq.m.|-|.(lim seq).|/2) =|.(lim seq).|/2+-(|.(lim seq).|/2-|.seq.m.|) by XCMPLX_1:143 .=|.(lim seq).|/2-(|.(lim seq).|/2-|.seq.m.|) by XCMPLX_0:def 8 .=|.(lim seq).|/2-|.(lim seq).|/2+|.seq.m.| by XCMPLX_1:37 .=0+|.seq.m.| by XCMPLX_1:14 .=|.seq.m.|; |.(lim seq).|-|.seq.m.|+(|.seq.m.|-|.(lim seq).|/2) =|.(lim seq).|-|.seq.m.|+|.seq.m.|-|.(lim seq).|/2 by XCMPLX_1:29 .=|.(lim seq).|-(|.seq.m.|-|.seq.m.|)-|.(lim seq).|/2 by XCMPLX_1:37 .=|.(lim seq).|-0-|.(lim seq).|/2 by XCMPLX_1:14 .=|.(lim seq).|/2+|.(lim seq).|/2 -|.(lim seq).|/2 by XCMPLX_1:66 .=|.(lim seq).|/2+(|.(lim seq).|/2 -|.(lim seq).|/2) by XCMPLX_1:29 .=|.(lim seq).|/2+0 by XCMPLX_1:14 .=|.(lim seq).|/2; hence |.(lim seq).|/2<|.seq.m.| by A7,A8,REAL_1:53; end;

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