:: Double Sequences and Limits
:: by Noboru Endou , Hiroyuki Okazaki and Yasunari Shidama
::
:: Received August 31, 2013
:: Copyright (c) 2013-2019 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, FUNCT_1, RELAT_1, XBOOLE_0, ARYTM_3, ARYTM_1,
XXREAL_0, CARD_1, TARSKI, NAT_1, ZFMISC_1, VALUED_0, ORDINAL2, SEQ_2,
XREAL_0, COMPLEX1, XCMPLX_0, XXREAL_2, FINSET_1, MEMBERED, MESFUNC9,
BHSP_3, SEQ_1, VALUED_1, DBLSEQ_1, FUNCOP_1, BINOP_1, BINOP_2, REAL_1;
notations ZFMISC_1, SUBSET_1, ORDINAL1, XXREAL_2, RELAT_1, PARTFUN1, RELSET_1,
FUNCT_1, FUNCT_2, NUMBERS, BINOP_1, XCMPLX_0, TARSKI, XBOOLE_0, FUNCOP_1,
FUNCT_3, FINSEQOP, MEMBERED, COMPLEX1, XXREAL_0, XREAL_0, SEQ_1, NAT_1,
MESFUNC9, VALUED_1, SEQ_2, FINSET_1, BINOP_2;
constructors TOPMETR, NAT_D, MESFUNC9, SEQ_2, COMSEQ_2, SEQ_4, REAL_1,
FINSEQOP, FUNCT_3, BINOP_2;
registrations ORDINAL1, XXREAL_0, XREAL_0, XBOOLE_0, NUMBERS, XCMPLX_0,
FINSET_1, SUBSET_1, MEMBERED, VALUED_0, XXREAL_2, RELSET_1, MEASURE6,
SEQ_4, FUNCT_1, FUNCT_2, VALUED_1, BINOP_2, NAT_1, SEQ_2, SEQ_1;
requirements SUBSET, REAL, BOOLE, NUMERALS, ARITHM;
begin
reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;
reserve rseq1,rseq2 for convergent Real_Sequence;
reserve n,m,N,M for Nat;
reserve e,r for Real;
:: Convergence in the Pringsheim sense
definition let Rseq;
attr Rseq is P-convergent means
:: DBLSEQ_1:def 1
ex p be Real st
for e be Real st 0=N & m>=N holds
|.Rseq.(n,m) - p.| < e;
end;
definition let Rseq;
assume Rseq is P-convergent;
func P-lim Rseq -> Real means
:: DBLSEQ_1:def 2
for e be Real st 0=N & m>=N holds
|. Rseq.(n,m) - it .| < e;
end;
definition let Rseq;
attr Rseq is convergent_in_cod1 means
:: DBLSEQ_1:def 3
for m be Element of NAT holds ProjMap2(Rseq,m) is convergent;
attr Rseq is convergent_in_cod2 means
:: DBLSEQ_1:def 4
for n be Element of NAT holds ProjMap1(Rseq,n) is convergent;
end;
definition let Rseq;
func lim_in_cod1 Rseq -> Function of NAT,REAL means
:: DBLSEQ_1:def 5
for m be Element of NAT holds it.m = lim ProjMap2(Rseq,m);
end;
definition let Rseq;
func lim_in_cod2 Rseq -> Function of NAT,REAL means
:: DBLSEQ_1:def 6
for n be Element of NAT holds it.n = lim ProjMap1(Rseq,n);
end;
definition let Rseq;
assume lim_in_cod1 Rseq is convergent;
func cod1_major_iterated_lim Rseq -> Real means
:: DBLSEQ_1:def 7
for e be Real st 0=M holds
|.(lim_in_cod1 Rseq).m - it.| < e;
end;
definition let Rseq;
assume lim_in_cod2 Rseq is convergent;
func cod2_major_iterated_lim Rseq -> Real means
:: DBLSEQ_1:def 8
for e be Real st 0=N holds
|.(lim_in_cod2 Rseq).n - it.| < e;
end;
definition
let Rseq be Function of [:NAT,NAT:],REAL;
attr Rseq is uniformly_convergent_in_cod1 means
:: DBLSEQ_1:def 9
Rseq is convergent_in_cod1 &
for e be Real st e>0
ex M be Nat st
for m be Nat st m>=M holds
for n be Nat holds
|. Rseq.(n,m) - (lim_in_cod1 Rseq).n .| < e;
end;
definition
let Rseq be Function of [:NAT,NAT:],REAL;
attr Rseq is uniformly_convergent_in_cod2 means
:: DBLSEQ_1:def 10
Rseq is convergent_in_cod2 &
for e be Real st e>0
ex N be Nat st
for n be Nat st n>=N holds
for m be Nat holds
|. Rseq.(n,m) - (lim_in_cod2 Rseq).m .| < e;
end;
definition let Rseq;
attr Rseq is non-decreasing means
:: DBLSEQ_1:def 11
for n1,m1,n2,m2 be Nat st
n1>=n2 & m1>=m2 holds Rseq.(n1,m1) >= Rseq.(n2,m2);
attr Rseq is non-increasing means
:: DBLSEQ_1:def 12
for n1,m1,n2,m2 be Nat st
n1>=n2 & m1>=m2 holds Rseq.(n1,m1) <= Rseq.(n2,m2);
end;
theorem :: DBLSEQ_1:1
for a,b,c be Real st a <= b & b <= c holds
|.b.| <= |.a.| or |.b.| <= |.c.|;
registration
cluster non-decreasing P-convergent -> bounded_below bounded_above
for Function of [:NAT,NAT:],REAL;
end;
registration
cluster non-increasing P-convergent -> bounded_below bounded_above
for Function of [:NAT,NAT:],REAL;
end;
registration
let r be Element of REAL;
cluster [:NAT,NAT:] --> r -> P-convergent convergent_in_cod1
convergent_in_cod2 for Function of [:NAT,NAT:],REAL;
end;
theorem :: DBLSEQ_1:2
for r be Element of REAL holds P-lim ([:NAT,NAT:] --> r) = r;
registration
cluster P-convergent convergent_in_cod1 convergent_in_cod2
for Function of [:NAT,NAT:],REAL;
end;
reserve Pseq for P-convergent Function of [:NAT,NAT:],REAL;
registration
let Pseq2 be P-convergent convergent_in_cod2 Function of [:NAT,NAT:],REAL;
cluster lim_in_cod2 Pseq2 -> convergent;
end;
theorem :: DBLSEQ_1:3
Rseq is P-convergent convergent_in_cod2 implies
P-lim Rseq = cod2_major_iterated_lim Rseq;
registration
let Pseq1 be P-convergent convergent_in_cod1 Function of [:NAT,NAT:],REAL;
cluster lim_in_cod1 Pseq1 -> convergent;
end;
theorem :: DBLSEQ_1:4
Rseq is P-convergent convergent_in_cod1 implies
P-lim Rseq = cod1_major_iterated_lim Rseq;
registration
cluster non-decreasing bounded_above ->
P-convergent convergent_in_cod1 convergent_in_cod2
for Function of [:NAT,NAT:],REAL;
end;
registration
cluster non-increasing bounded_below ->
P-convergent convergent_in_cod1 convergent_in_cod2
for Function of [:NAT,NAT:],REAL;
end;
theorem :: DBLSEQ_1:5
Rseq is uniformly_convergent_in_cod1 & lim_in_cod1 Rseq is convergent implies
Rseq is P-convergent & P-lim Rseq = cod1_major_iterated_lim Rseq;
theorem :: DBLSEQ_1:6
Rseq is uniformly_convergent_in_cod2 & lim_in_cod2 Rseq is convergent implies
Rseq is P-convergent & P-lim Rseq = cod2_major_iterated_lim Rseq;
definition
let Rseq;
attr Rseq is Cauchy means
:: DBLSEQ_1:def 13
for e be Real st e>0
ex N be Nat st
for n1,n2,m1,m2 be Nat st N<=n1 & n1<=n2 & N<=m1 & m1<=m2
holds |.Rseq.(n2,m2) - Rseq.(n1,m1).| < e;
end;
theorem :: DBLSEQ_1:7
Rseq is P-convergent iff Rseq is Cauchy;
theorem :: DBLSEQ_1:8
for Rseq be Function of [:NAT,NAT:],REAL st
Rseq is non-decreasing or Rseq is non-increasing holds
Rseq is P-convergent iff Rseq is bounded_below bounded_above;
definition
let X,Y be non empty set;
let H be BinOp of Y;
let f,g be Function of X,Y;
redefine func H*(f,g) -> Function of [:X,X:],Y;
end;
theorem :: DBLSEQ_1:9
multreal*(rseq1,rseq2) is convergent_in_cod1 convergent_in_cod2
& lim_in_cod1 (multreal*(rseq1,rseq2)) is convergent
& cod1_major_iterated_lim (multreal*(rseq1,rseq2)) = lim rseq1 * lim rseq2
& lim_in_cod2 (multreal*(rseq1,rseq2)) is convergent
& cod2_major_iterated_lim (multreal*(rseq1,rseq2)) = lim rseq1 * lim rseq2
& multreal*(rseq1,rseq2) is P-convergent
& P-lim (multreal*(rseq1,rseq2)) = lim rseq1 * lim rseq2;
theorem :: DBLSEQ_1:10
addreal*(rseq1,rseq2) is convergent_in_cod1 convergent_in_cod2
& lim_in_cod1 (addreal*(rseq1,rseq2)) is convergent
& cod1_major_iterated_lim (addreal*(rseq1,rseq2)) = lim rseq1 + lim rseq2
& lim_in_cod2 (addreal*(rseq1,rseq2)) is convergent
& cod2_major_iterated_lim (addreal*(rseq1,rseq2)) = lim rseq1 + lim rseq2
& addreal*(rseq1,rseq2) is P-convergent
& P-lim (addreal*(rseq1,rseq2)) = lim rseq1 + lim rseq2;
theorem :: DBLSEQ_1:11
Rseq1 is P-convergent & Rseq2 is P-convergent implies
Rseq1+Rseq2 is P-convergent & P-lim(Rseq1+Rseq2) = P-lim Rseq1 + P-lim Rseq2;
theorem :: DBLSEQ_1:12
Rseq1 is P-convergent & Rseq2 is P-convergent implies
Rseq1-Rseq2 is P-convergent & P-lim(Rseq1-Rseq2) = P-lim Rseq1 - P-lim Rseq2;
theorem :: DBLSEQ_1:13
for Rseq be Function of [:NAT,NAT:],REAL, r be Real st
Rseq is P-convergent holds
r(#)Rseq is P-convergent & P-lim (r(#)Rseq) = r * P-lim Rseq;
theorem :: DBLSEQ_1:14
Rseq is P-convergent &(for n,m be Nat holds Rseq.(n,m) >= r)
implies P-lim Rseq >= r;
theorem :: DBLSEQ_1:15
Rseq1 is P-convergent & Rseq2 is P-convergent &
(for n,m be Nat holds Rseq1.(n,m) <= Rseq2.(n,m)) implies
P-lim Rseq1 <= P-lim Rseq2;
theorem :: DBLSEQ_1:16
Rseq1 is P-convergent & Rseq2 is P-convergent & P-lim Rseq1 = P-lim Rseq2 &
(for n,m be Nat holds
Rseq1.(n,m) <= Rseq.(n,m) & Rseq.(n,m) <= Rseq2.(n,m) )
implies Rseq is P-convergent & P-lim Rseq = P-lim Rseq1;
definition
let X be non empty set, seq be Function of [:NAT,NAT:],X;
mode subsequence of seq -> Function of [:NAT,NAT:],X means
:: DBLSEQ_1:def 14
ex N,M be increasing sequence of NAT st
for n,m be Nat holds it.(n,m) = seq.(N.n,M.m);
end;
registration
let Pseq;
cluster -> P-convergent for subsequence of Pseq;
end;
theorem :: DBLSEQ_1:17
for Psubseq be subsequence of Pseq holds
P-lim Psubseq = P-lim Pseq;
registration
let Rseq be convergent_in_cod1 Function of [:NAT,NAT:],REAL;
cluster -> convergent_in_cod1 for subsequence of Rseq;
end;
theorem :: DBLSEQ_1:18
for Rseq1 be subsequence of Rseq st
Rseq is convergent_in_cod1 & lim_in_cod1 Rseq is convergent holds
lim_in_cod1 Rseq1 is convergent &
cod1_major_iterated_lim Rseq1 = cod1_major_iterated_lim Rseq;
registration
let Rseq be convergent_in_cod2 Function of [:NAT,NAT:],REAL;
cluster -> convergent_in_cod2 for subsequence of Rseq;
end;
theorem :: DBLSEQ_1:19
for Rseq1 be subsequence of Rseq st
Rseq is convergent_in_cod2 & lim_in_cod2 Rseq is convergent holds
lim_in_cod2 Rseq1 is convergent &
cod2_major_iterated_lim Rseq1 = cod2_major_iterated_lim Rseq;