reserve X for MetrSpace,
  x,y,z for Element of X,
  A for non empty set,
  G for Function of [:A,A:],REAL,
  f for Function,
  k,n,m,m1,m2 for Nat,
  q,r for Real;
reserve X for non empty MetrSpace,
  x,y for Element of X,
  V for Subset of X,
  S,S1,T for sequence of X,
  Nseq for increasing sequence of NAT;

theorem Th23:
  S is constant implies S is convergent
proof
  assume S is constant;
  then consider x such that
A1: for n being Nat holds S.n =x by VALUED_0:def 18;
  take x;
  let r such that
A2: 0 < r;
  take k = 0;
  now
    let n such that
    k <= n;
    dist(S.n,x) = dist(x,x) by A1
      .= 0 by METRIC_1:1;
    hence dist(S.n,x) < r by A2;
  end;
  hence for n st k <= n holds dist(S.n,x) < r;
end;
