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 Element of NAT st
for m being Element of NAT st n <= m holds
||.((seq . m) - g1).|| < p by A2, NORMSP_1:def 6;
take g1 ; :: according to NORMSP_1:def 6 :: thesis: for b₁ being Element of REAL holds
( b₁ <= 0 or ex b₂ being Element of NAT st
for b₃ being Element of NAT holds
( not b₂ <= b₃ or not b₁ <= ||.((seq1 . b₃) - g1).|| ) )
let p be Real; :: thesis: ( p <= 0 or ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= ||.((seq1 . b₂) - g1).|| ) )
assume 0 < p ; :: thesis: ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= ||.((seq1 . b₂) - g1).|| )
then consider n1 being Element of NAT such that
A4: for m being Element of NAT st n1 <= m holds
||.((seq . m) - g1).|| < p by A3;
take n = n1; :: thesis: for b₁ being Element of NAT holds
( not n <= b₁ or not p <= ||.((seq1 . b₁) - g1).|| )
let m be Element of NAT ; :: thesis: ( not n <= m or not p <= ||.((seq1 . m) - g1).|| )
assume A5: n <= m ; :: thesis: not p <= ||.((seq1 . m) - g1).||
consider Nseq being increasing sequence of NAT such that
A6: seq1 = seq * Nseq by A1, VALUED_0:def 17;
m <= Nseq . m by SEQM_3:14;
then A7: n <= Nseq . m by A5, XXREAL_0:2;
seq1 . m = seq . (Nseq . m) by A6, FUNCT_2:15;
hence not p <= ||.((seq1 . m) - g1).|| by A4, A7; :: thesis: verum