let X be RealNormSpace; :: thesis: for seq, seq1 being sequence of X st seq1 is subsequence of seq & seq is convergent holds

seq1 is convergent

let seq, seq1 be sequence of X; :: thesis: ( seq1 is subsequence of seq & seq is convergent implies seq1 is convergent )

assume that

A1: seq1 is subsequence of seq and

A2: seq is convergent ; :: thesis: seq1 is convergent

consider g1 being Element of X such that

A3: for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

||.((seq . m) - g1).|| < p by A2;

take g1 ; :: according to NORMSP_1:def 6 :: thesis: for b_{1} being object holds

( b_{1} <= 0 or ex b_{2} being set st

for b_{3} being set holds

( not b_{2} <= b_{3} or not b_{1} <= ||.((seq1 . b_{3}) - g1).|| ) )

let p be Real; :: thesis: ( p <= 0 or ex b_{1} being set st

for b_{2} being set holds

( not b_{1} <= b_{2} or not p <= ||.((seq1 . b_{2}) - g1).|| ) )

assume 0 < p ; :: thesis: ex b_{1} being set st

for b_{2} being set holds

( not b_{1} <= b_{2} or not p <= ||.((seq1 . b_{2}) - g1).|| )

then consider n1 being Nat such that

A4: for m being Nat st n1 <= m holds

||.((seq . m) - g1).|| < p by A3;

take n = n1; :: thesis: for b_{1} being set holds

( not n <= b_{1} or not p <= ||.((seq1 . b_{1}) - g1).|| )

let m be Nat; :: thesis: ( not n <= m or not p <= ||.((seq1 . m) - g1).|| )

assume A5: n <= m ; :: thesis: not p <= ||.((seq1 . m) - g1).||

A6: m in NAT by ORDINAL1:def 12;

consider Nseq being increasing sequence of NAT such that

A7: seq1 = seq * Nseq by A1, VALUED_0:def 17;

m <= Nseq . m by SEQM_3:14;

then A8: n <= Nseq . m by A5, XXREAL_0:2;

seq1 . m = seq . (Nseq . m) by A7, FUNCT_2:15, A6;

hence not p <= ||.((seq1 . m) - g1).|| by A4, A8; :: thesis: verum

seq1 is convergent

let seq, seq1 be sequence of X; :: thesis: ( seq1 is subsequence of seq & seq is convergent implies seq1 is convergent )

assume that

A1: seq1 is subsequence of seq and

A2: seq is convergent ; :: thesis: seq1 is convergent

consider g1 being Element of X such that

A3: for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

||.((seq . m) - g1).|| < p by A2;

take g1 ; :: according to NORMSP_1:def 6 :: thesis: for b

( b

for b

( not b

let p be Real; :: thesis: ( p <= 0 or ex b

for b

( not b

assume 0 < p ; :: thesis: ex b

for b

( not b

then consider n1 being Nat such that

A4: for m being Nat st n1 <= m holds

||.((seq . m) - g1).|| < p by A3;

take n = n1; :: thesis: for b

( not n <= b

let m be Nat; :: thesis: ( not n <= m or not p <= ||.((seq1 . m) - g1).|| )

assume A5: n <= m ; :: thesis: not p <= ||.((seq1 . m) - g1).||

A6: m in NAT by ORDINAL1:def 12;

consider Nseq being increasing sequence of NAT such that

A7: seq1 = seq * Nseq by A1, VALUED_0:def 17;

m <= Nseq . m by SEQM_3:14;

then A8: n <= Nseq . m by A5, XXREAL_0:2;

seq1 . m = seq . (Nseq . m) by A7, FUNCT_2:15, A6;

hence not p <= ||.((seq1 . m) - g1).|| by A4, A8; :: thesis: verum