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 Th85:
  seq is Cauchy & seq1 is subsequence of seq implies seq1 is Cauchy
proof
  assume that
A1: seq is Cauchy and
A2: seq1 is subsequence of seq;
  consider Nseq such that
A3: seq1 = seq * Nseq by A2,VALUED_0:def 17;
  now
    let r;
    assume r > 0;
    then consider l be Nat such that
A4: for n, m st n >= l & m >= l holds dist((seq.n), (seq.m)) < r by A1;
    take k = l;
    let n, m such that
A5: n >= k and
A6: m >= k;
A7: n in NAT & m in NAT by ORDINAL1:def 12;
    Nseq.n >= n by SEQM_3:14;
    then
A8: Nseq.n >= k by A5,XXREAL_0:2;
    Nseq.m >= m by SEQM_3:14;
    then
A9: Nseq.m >= k by A6,XXREAL_0:2;
    seq1.n = seq.(Nseq.n) & seq1.m = seq.(Nseq.m) by A3,FUNCT_2:15,A7;
    hence dist((seq1.n), (seq1.m)) < r by A4,A8,A9;
  end;
  hence thesis;
end;
