reserve k, k1, n, n1, m for Nat;
reserve X, y for set;
reserve p for Real;
reserve r for Real;
reserve a, a1, a2, b, b1, b2, x, x0, z, z0 for Complex;
reserve s1, s3, seq, seq1 for Complex_Sequence;
reserve Y for Subset of COMPLEX;
reserve f, f1, f2 for PartFunc of COMPLEX,COMPLEX;
reserve Nseq for increasing sequence of NAT;

theorem Th2:
  0 < r implies InvShift(r) is convergent
proof
  assume
A1: 0 < r;
  set seq = InvShift(r);
  take g = 0c;
  let p;
  assume
A2: 0 < p;
  consider k1 such that
A3: p" < k1 by SEQ_4:3;
  take n = k1;
  let m;
  assume n <= m;
  then n+r <= m+r by XREAL_1:6;
  then
A4: 1/(m+r) <= 1/(n+r) by A1,XREAL_1:118;
A5: seq.m = 1/(m+r) by Def2;
  (p") + 0 < k1+r by A1,A3,XREAL_1:8;
  then 1/(k1+r) < 1/p" by A2,XREAL_1:76;
  then 1/(m+r) < p by A4,XXREAL_0:2;
  hence |.seq.m-g.| < p by A1,A5,ABSVALUE:def 1;
end;
