reserve n,n1,n2,m for Nat;
reserve r,g1,g2,g,g9 for Complex;
reserve R,R2 for Real;
reserve s,s9,s1 for Complex_Sequence;

theorem Th9:
  (ex g st for n being Nat holds s.n = g) implies s is convergent
proof
  reconsider zz=0 as Nat;
  given g such that
A1: for n being Nat holds s.n = g;
  take g;
  now
    let p be Real such that
A2: 0<p;
    take k = zz;
    let n such that
    k<=n;
    s.n = g by A1;
    hence |.s.n - g.| < p by A2,COMPLEX1:44;
  end;
  hence thesis;
end;
