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 Th84:
  seq1 is subsequence of seq & seq is convergent implies lim seq1= lim seq
proof
  assume that
A1: seq1 is subsequence of seq and
A2: seq is convergent;
  consider Nseq such that
A3: seq1 = seq * Nseq by A1,VALUED_0:def 17;
A4: now
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A5: for n st n >= m1 holds dist((seq.n), (lim seq)) < r by A2,Def2;
    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 dist((seq1.n), (lim seq)) < r by A5,A7;
  end;
  seq1 is convergent by A1,A2,Th83;
  hence thesis by A4,Def2;
end;
