let k be Element of NAT ; :: thesis: for seq being ExtREAL_sequence st seq is convergent_to_finite_number holds
( seq ^\ k is convergent_to_finite_number & seq ^\ k is convergent & lim seq = lim (seq ^\ k) )
let seq be ExtREAL_sequence; :: thesis: ( seq is convergent_to_finite_number implies ( seq ^\ k is convergent_to_finite_number & seq ^\ k is convergent & lim seq = lim (seq ^\ k) ) )
set seq0 = seq ^\ k;
assume A1: seq is convergent_to_finite_number ; :: thesis: ( seq ^\ k is convergent_to_finite_number & seq ^\ k is convergent & lim seq = lim (seq ^\ k) )
then A2: ( not lim seq = +infty or not seq is convergent_to_+infty ) by MESFUNC5:56;
A3: ( not lim seq = -infty or not seq is convergent_to_-infty ) by A1, MESFUNC5:57;
seq is convergent by A1, MESFUNC5:def 11;
then A4: ex g being real number st
( lim seq = g & ( for p being real number st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - (lim seq)).| < p ) & seq is convergent_to_finite_number ) by A2, A3, MESFUNC5:def 12;
then consider g being real number such that
A5: lim seq = g ;
A6: for p being real number st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.(((seq ^\ k) . m) - (lim seq)).| < p

proof

let p be real number ; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
|.(((seq ^\ k) . m) - (lim seq)).| < p )
assume 0 < p ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.(((seq ^\ k) . m) - (lim seq)).| < p
then consider n being Nat such that
A7: for m being Nat st n <= m holds
|.((seq . m) - (lim seq)).| < p by A4;
take n ; :: thesis: for m being Nat st n <= m holds
|.(((seq ^\ k) . m) - (lim seq)).| < p

hereby :: thesis: verum

let m be Nat; :: thesis: ( n <= m implies |.(((seq ^\ k) . m) - (lim seq)).| < p )
assume A8: n <= m ; :: thesis: |.(((seq ^\ k) . m) - (lim seq)).| < p
m <= m + k by NAT_1:11;
then n <= m + k by A8, XXREAL_0:2;
then |.((seq . (m + k)) - (lim seq)).| < p by A7;
hence |.(((seq ^\ k) . m) - (lim seq)).| < p by NAT_1:def 3; :: thesis: verum

end;

end;

lim seq = R_EAL g by A5;
hence A9: seq ^\ k is convergent_to_finite_number by A6, MESFUNC5:def 8; :: thesis: ( seq ^\ k is convergent & lim seq = lim (seq ^\ k) )
hence seq ^\ k is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)
hence lim seq = lim (seq ^\ k) by A5, A6, A9, MESFUNC5:def 12; :: thesis: verum