let r be Real; :: thesis: for seq being Real_Sequence st 0 <= r & ( for n being Nat holds seq . n = 1 / ((n * n) + r) ) holds
seq is convergent
let seq be Real_Sequence; :: thesis: ( 0 <= r & ( for n being Nat holds seq . n = 1 / ((n * n) + r) ) implies seq is convergent )
assume that
A1: 0 <= r and
A2: for n being Nat holds seq . n = 1 / ((n * n) + r) ; :: thesis: seq is convergent
take 0 ; :: according to SEQ_2:def 6 :: thesis: for b₁ being set holds
( b₁ <= 0 or ex b₂ being set st
for b₃ being set holds
( not b₂ <= b₃ or not b₁ <= |.((seq . b₃) - 0).| ) )
let p be Real; :: thesis: ( p <= 0 or ex b₁ being set st
for b₂ being set holds
( not b₁ <= b₂ or not p <= |.((seq . b₂) - 0).| ) )
consider k1 being Nat such that
A3: p " < k1 by Th3;
assume A4: 0 < p ; :: thesis: ex b₁ being set st
for b₂ being set holds
( not b₁ <= b₂ or not p <= |.((seq . b₂) - 0).| )
then A5: k1 > 0 by A3;
then k1 >= 1 + 0 by NAT_1:13;
then k1 <= k1 * k1 by XREAL_1:151;
then A6: k1 + r <= (k1 * k1) + r by XREAL_1:6;
take n = k1; :: thesis: for b₁ being set holds
( not n <= b₁ or not p <= |.((seq . b₁) - 0).| )
let m be Nat; :: thesis: ( not n <= m or not p <= |.((seq . m) - 0).| )
assume A7: n <= m ; :: thesis: not p <= |.((seq . m) - 0).|
n * n <= m * m by A7, XREAL_1:66;
then A8: (n * n) + r <= (m * m) + r by XREAL_1:6;
(p ") + 0 < k1 + r by A1, A3, XREAL_1:8;
then (p ") + 0 < (k1 * k1) + r by A6, XXREAL_0:2;
then 1 / ((k1 * k1) + r) < 1 / (p ") by A4, XREAL_1:76;
then A9: 1 / ((k1 * k1) + r) < 1 * ((p ") ") by XCMPLX_0:def 9;
0 < n ^2 by A5;
then 1 / ((m * m) + r) <= 1 / ((n * n) + r) by A1, A8, XREAL_1:118;
then A10: 1 / ((m * m) + r) < p by A9, XXREAL_0:2;
seq . m = 1 / ((m * m) + r) by A2;
hence not p <= |.((seq . m) - 0).| by A1, A10, ABSVALUE:def 1; :: thesis: verum