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

theorem Th7:
  seq is convergent & lim seq=x0 & 0<g implies ex k st for n st k<=
  n holds x0-g<seq.n & seq.n<x0+g
proof
  assume that
A1: seq is convergent and
A2: lim seq=x0 and
A3: 0<g;
  x0-g<lim seq by A2,A3,Lm1;
  then consider k1 be Nat such that
A4: for n being Nat st k1<=n holds x0-g<seq.n by A1,LIMFUNC2:1;
  lim seq<x0+g by A2,A3,Lm1;
  then consider k2 be Nat such that
A5: for n being Nat st k2<= n holds seq.n<x0+g by A1,LIMFUNC2:2;
   reconsider k=max(k1,k2)as Element of NAT by ORDINAL1:def 12;
  take k;
  let n;
  assume
A6: k<=n;
  k1<=k by XXREAL_0:25;
  then k1<=n by A6,XXREAL_0:2;
  hence x0-g<seq.n by A4;
  k2<=k by XXREAL_0:25;
  then k2<=n by A6,XXREAL_0:2;
  hence thesis by A5;
end;
