reserve r for Real,
  X for set,
  f, g, h for real-valued Function;

theorem
  for f being Real_Sequence for a,r being positive Real st r <
1 & for n being Nat holds |.f.n-f.(n+1).| <= a*(r to_power n) holds
  lim f >= f.0-a/(1-r) & lim f <= f.0+a/(1-r)
proof
  let f be Real_Sequence;
  let a,r be positive Real;
  assume
A1: r < 1;
A2: r to_power 0 = 1 by POWER:24;
  assume for n being Nat holds |.f.n-f.(n+1).| <= a*(r to_power n);
  then |.(lim f)-(f.0).| <= a*1/(1-r) by A1,A2,Th7;
  hence thesis by Th1;
end;
