reserve N for Nat;
reserve n,m,n1,n2 for Nat;
reserve q,r,r1,r2 for Real;
reserve x,y for set;
reserve w,w1,w2,g,g1,g2 for Point of TOP-REAL N;
reserve seq,seq1,seq2,seq3,seq9 for Real_Sequence of N;

theorem
  seq is convergent implies ((lim seq) <> 0.TOP-REAL N implies ex n st
  for m st n<=m holds |.(lim seq).|/2 < |.seq.m.|)
proof
  assume that
A1: seq is convergent and
A2: (lim seq)<>0.TOP-REAL N;
  0 <> |.(lim seq).| by A2,Th24;
  then 0<|.(lim seq).|/2 by XREAL_1:215;
  then consider n1 such that
A3: for m st n1<=m holds |.seq.m-(lim seq).|<|.(lim seq).|/2 by A1,Def9;
  take n=n1;
  let m;
  assume n<=m;
  then
A4: |.seq.m-(lim seq).|<|.(lim seq).|/2 by A3;
  |.(lim seq).|-|.seq.m.|<=|.(lim seq)-seq.m.| & |.(lim seq)-seq.m.|=|.seq
  .m-( lim seq).| by Th27,Th32;
  then
A5: |.(lim seq).|-|.seq.m.|<|.(lim seq).|/2 by A4,XXREAL_0:2;
  |.(lim seq).|/2+(|.seq.m.|-|.(lim seq).|/2) =|.seq.m.| & |.(lim seq).|-
  |.seq .m.|+(|.seq.m.|-|.(lim seq).|/2) =|.(lim seq).|/2;
  hence thesis by A5,XREAL_1:6;
end;
