reserve r,r1,r2,g,g1,g2,x0,t for Real;
reserve n,k for Nat;
reserve seq for Real_Sequence;
reserve f,f1,f2 for PartFunc of REAL,REAL;

theorem Th2:
  seq is convergent & lim seq<r implies ex n st for k st n<=k holds seq.k<r
proof
  assume that
A1: seq is convergent and
A2: lim seq<r;
  reconsider rr = r as Element of REAL by XREAL_0:def 1;
  set s = seq_const r;
A3: s-seq is convergent by A1;
  s.0=r by SEQ_1:57;
  then lim s=r by SEQ_4:25;
  then lim(s-seq)=r-lim seq by A1,SEQ_2:12;
  then consider n such that
A4: for k st n<=k holds 0<(s-seq).k by A2,A3,LIMFUNC1:4,XREAL_1:50;
  take n;
  let k;
  assume n<=k;
  then 0<(s-seq).k by A4;
  then 0<s.k-seq.k by RFUNCT_2:1;
  then 0<r-seq.k by SEQ_1:57;
  then 0+seq.k<r by XREAL_1:20;
  hence thesis;
end;
