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 Th1:
  seq is convergent & r<lim seq implies ex n st for k st n<=k holds r<seq.k
proof
  assume that
A1: seq is convergent and
A2: r<lim seq;
  reconsider rr = r as Element of REAL by XREAL_0:def 1;
  set s = seq_const r;
A3: seq-s is convergent by A1;
  s.0=r by SEQ_1:57;
  then lim s=r by SEQ_4:25;
  then lim(seq-s)=lim seq -r by A1,SEQ_2:12;
  then consider n such that
A4: for k st n<= k holds 0<(seq-s).k by A2,A3,LIMFUNC1:4,XREAL_1:50;
  take n;
  let k;
  assume n<=k;
  then 0<(seq-s).k by A4;
  then 0<seq.k-s.k by RFUNCT_2:1;
  then 0<seq.k-r by SEQ_1:57;
  then 0+r<seq .k by XREAL_1:20;
  hence thesis;
end;
