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 Th22:
  for s being convergent Complex_Sequence st lim s <> 0c
   ex n st for m st n <=m holds |.(lim s).|/2<|.s.m.|
proof
  let s be convergent Complex_Sequence such that
A1: (lim s)<>0c;
  0<|.(lim s).| by A1,COMPLEX1:47;
  then 0<|.(lim s).|/2;
  then consider n1 such that
A2: for m st n1<=m holds |.s.m-(lim s).|<|.(lim s).|/2 by Def6;
  take n=n1;
  let m;
  assume n<=m;
  then
A3: |.s.m-(lim s).|<|.(lim s).|/2 by A2;
A4: |.(lim s)-s.m.|=|.-(s.m-(lim s)).| .=|.s.m-(lim s).| by COMPLEX1:52;
A5: |.(lim s).|/2+(|.s.m.|- |.(lim s).|/2) =|.s.m.| & |.(lim s).|- |.s.m.|+
  (|.s. m.|- |.(lim s).|/2) =|.(lim s).|/2;
  |.(lim s).|- |.s.m.|<=|.(lim s)-s.m.| by COMPLEX1:59;
  then |.(lim s).|- |.s.m.|<|.(lim s).|/2 by A3,A4,XXREAL_0:2;
  hence thesis by A5,XREAL_1:6;
end;
