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 Th10:
  for g st for n being Nat holds s.n = g holds lim s = g
proof
  let g;
  assume
A1: for n being Nat holds s.n = g;
A2: now
    let p be Real such that
A3: 0<p;
    reconsider zz=0 as Nat;
    take k = zz;
    let n such that
    k<=n;
    s.n = g by A1;
    hence |.s.n - g.| < p by A3,COMPLEX1:44;
  end;
  s is convergent by A1,Th9;
  hence thesis by A2,Def6;
end;
