let r be Real; :: thesis: ( 0 < r implies InvShift r is convergent )
assume A1: 0 < r ; :: thesis: InvShift r is convergent
set seq = InvShift r;
take g = 0c ; :: according to COMSEQ_2:def 5 :: thesis: for b₁ being object holds
( b₁ <= 0 or ex b₂ being set st
for b₃ being set holds
( not b₂ <= b₃ or not b₁ <= |.(((InvShift r) . b₃) - g).| ) )
let p be Real; :: thesis: ( p <= 0 or ex b₁ being set st
for b₂ being set holds
( not b₁ <= b₂ or not p <= |.(((InvShift r) . b₂) - g).| ) )
assume A2: 0 < p ; :: thesis: ex b₁ being set st
for b₂ being set holds
( not b₁ <= b₂ or not p <= |.(((InvShift r) . b₂) - g).| )
consider k1 being Nat such that
A3: p " < k1 by SEQ_4:3;
take n = k1; :: thesis: for b₁ being set holds
( not n <= b₁ or not p <= |.(((InvShift r) . b₁) - g).| )
let m be Nat; :: thesis: ( not n <= m or not p <= |.(((InvShift r) . m) - g).| )
assume n <= m ; :: thesis: not p <= |.(((InvShift r) . m) - g).|
then n + r <= m + r by XREAL_1:6;
then A4: 1 / (m + r) <= 1 / (n + r) by A1, XREAL_1:118;
A5: (InvShift r) . 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 |.(((InvShift r) . m) - g).| < p by A1, A5, ABSVALUE:def 1; :: thesis: verum