reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem Th83:
  seq is convergent & seq1 is subsequence of seq implies seq1 is convergent
proof
  assume that
A1: seq is convergent and
A2: seq1 is subsequence of seq;
  consider Nseq such that
A3: seq1 = seq * Nseq by A2,VALUED_0:def 17;
  consider g1 such that
A4: for r st r > 0 ex m st for n st n >= m holds ||.(seq.n) - g1.|| < r
  by A1,Th9;
  now
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A5: for n st n >= m1 holds ||.(seq.n) - g1.|| < r by A4;
    take m = m1;
    let n such that
A6: n >= m;
    Nseq.n >= n by SEQM_3:14;
    then
A7: Nseq.n >= m by A6,XXREAL_0:2;
    n in NAT by ORDINAL1:def 12;
    then seq1.n = seq.(Nseq.n) by A3,FUNCT_2:15;
    hence ||.(seq1.n) - g1.|| < r by A5,A7;
  end;
  hence thesis by Th9;
end;
