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 Th23:
  seq is convergent & lim seq = g implies dist(seq,g) is convergent
proof
  assume
A1: seq is convergent & lim seq = g;
  now
    let r be Real;
    assume
A2: r > 0;
    consider m1 be Nat such that
A3: for n st n >= m1 holds dist((seq.n) , g) < r by A1,A2,Def2;
    take k = m1;
    now
      let n;
      dist((seq.n) , g) >= 0 by CSSPACE:53;
      then
A4:   |.(dist((seq.n) , g) - 0).| = dist((seq.n) , g) by ABSVALUE:def 1;
      assume n >= k;
      then dist((seq.n) , g) < r by A3;
      hence |.(dist(seq, g).n - 0).| < r by A4,Def4;
    end;
    hence for n st k <= n holds |.(dist(seq, g).n - 0).| < r;
  end;
  hence thesis by SEQ_2:def 6;
end;
