reserve n,n1,n2,m for Nat,
  r,r1,r2,p,g1,g2,g for Real,
  seq,seq9,seq1 for Real_Sequence,
  y for set;

theorem Th13:
  seq is convergent implies seq is bounded
proof
  assume seq is convergent;
  then consider g such that
A1: for p st 0<p ex n st for m st n<=m holds |.seq.m-g.|<p;
  consider n1 such that
A2: for m st n1<=m holds |.seq.m-g.|<1 by A1;
A3: now
    take r=|.g.|+1;
    0+0<r by COMPLEX1:46,XREAL_1:8;
    hence 0<r;
    let m;
    assume n1<=m;
    then
A4: |.seq.m-g.|<1 by A2;
    |.seq.m.|-|.g.|<=|.seq.m-g.| by COMPLEX1:59;
    then
A5: |.seq.m.|-|.g.|<1 by A4,XXREAL_0:2;
    |.seq.m.|-|.g.|+|.g.|=|.seq.m.|;
    hence |.seq.m.|<r by A5,XREAL_1:6;
  end;
  now consider r1 such that
A6: 0<r1 and
A7: for m st n1<=m holds |.seq.m.|<r1 by A3;
    consider r2 such that
A8: 0<r2 and
A9: for m st m<=n1 holds |.seq.m.|<r2 by Th4;
    take r=r1+r2;
    thus 0<r by A6,A8;
A10: r1+0<r by A8,XREAL_1:8;
A11: 0+r2<r by A6,XREAL_1:8;
    let m;
A12: now
      assume n1<=m;
      then |.seq.m.|<r1 by A7;
      hence |.seq.m.|<r by A10,XXREAL_0:2;
    end;
    now
      assume m<=n1;
      then |.seq.m.|<r2 by A9;
      hence |.seq.m.|<r by A11,XXREAL_0:2;
    end;
    hence |.seq.m.|<r by A12;
  end;
  hence thesis by Th3;
end;
