let S be RealNormSpace; :: thesis: for seq being sequence of S
for x0 being Point of S
for r being Real st 0 < r & ( for n being Nat holds seq . n = (1 / (n + r)) * x0 ) holds
seq is convergent

let seq be sequence of S; :: thesis: for x0 being Point of S
for r being Real st 0 < r & ( for n being Nat holds seq . n = (1 / (n + r)) * x0 ) holds
seq is convergent

let x0 be Point of S; :: thesis: for r being Real st 0 < r & ( for n being Nat holds seq . n = (1 / (n + r)) * x0 ) holds
seq is convergent

let r be Real; :: thesis: ( 0 < r & ( for n being Nat holds seq . n = (1 / (n + r)) * x0 ) implies seq is convergent )
assume that
A1: 0 < r and
A2: for n being Nat holds seq . n = (1 / (n + r)) * x0 ; :: thesis: seq is convergent
take g = 0. S; :: according to NORMSP_1:def 6 :: 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 <= ||.((seq . 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 <= ||.((seq . b2) - g).|| ) )

assume A3: 0 < p ; :: thesis: ex b1 being set st
for b2 being set holds
( not b1 <= b2 or not p <= ||.((seq . b2) - g).|| )

ex pp being Real st
( pp > 0 & pp * ||.x0.|| < p )
proof
take pp = p / (||.x0.|| + 1); :: thesis: ( pp > 0 & pp * ||.x0.|| < p )
A4: ( ||.x0.|| + 0 < ||.x0.|| + 1 & 0 <= ||.x0.|| ) by ;
A5: ||.x0.|| + 1 > 0 + 0 by ;
then 0 < p / (||.x0.|| + 1) by ;
then pp * ||.x0.|| < pp * (||.x0.|| + 1) by ;
hence ( pp > 0 & pp * ||.x0.|| < p ) by ; :: thesis: verum
end;
then consider pp being Real such that
A6: pp > 0 and
A7: pp * ||.x0.|| < p ;
consider k1 being Nat such that
A8: pp " < k1 by SEQ_4:3;
(pp ") + 0 < k1 + r by ;
then 1 / (k1 + r) < 1 / (pp ") by ;
then A9: 1 / (k1 + r) < 1 * ((pp ") ") by XCMPLX_0:def 9;
reconsider k1 = k1 as Element of NAT by ORDINAL1:def 12;
take n = k1; :: thesis: for b1 being set holds
( not n <= b1 or not p <= ||.((seq . b1) - g).|| )

let m be Nat; :: thesis: ( not n <= m or not p <= ||.((seq . m) - g).|| )
assume n <= m ; :: thesis: not p <= ||.((seq . m) - g).||
then A10: n + r <= m + r by XREAL_1:6;
A11: 0 <= ||.x0.|| by NORMSP_1:4;
1 / (m + r) <= 1 / (n + r) by ;
then 1 / (m + r) < pp by ;
then A12: (1 / (m + r)) * ||.x0.|| <= pp * ||.x0.|| by ;
||.((seq . m) - g).|| = ||.(((1 / (m + r)) * x0) - (0. S)).|| by A2
.= ||.((1 / (m + r)) * x0).|| by RLVECT_1:13
.= |.(1 / (m + r)).| * ||.x0.|| by NORMSP_1:def 1
.= (1 / (m + r)) * ||.x0.|| by ;
hence not p <= ||.((seq . m) - g).|| by ; :: thesis: verum