let k be Nat; :: thesis: for seq being ExtREAL_sequence st seq ^\ k is convergent_to_finite_number holds

( seq is convergent_to_finite_number & seq is convergent & lim seq = lim (seq ^\ k) )

let seq be ExtREAL_sequence; :: thesis: ( seq ^\ k is convergent_to_finite_number implies ( seq is convergent_to_finite_number & seq is convergent & lim seq = lim (seq ^\ k) ) )

set seq0 = seq ^\ k;

assume A1: seq ^\ k is convergent_to_finite_number ; :: thesis: ( seq is convergent_to_finite_number & seq is convergent & lim seq = lim (seq ^\ k) )

then A2: ( not lim (seq ^\ k) = +infty or not seq ^\ k is convergent_to_+infty ) by MESFUNC5:50;

A3: ( not lim (seq ^\ k) = -infty or not seq ^\ k is convergent_to_-infty ) by A1, MESFUNC5:51;

seq ^\ k is convergent by A1, MESFUNC5:def 11;

then A4: ex g being Real st

( lim (seq ^\ k) = g & ( for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

|.(((seq ^\ k) . m) - (lim (seq ^\ k))).| < p ) & seq ^\ k is convergent_to_finite_number ) by A2, A3, MESFUNC5:def 12;

then consider g being Real such that

A5: lim (seq ^\ k) = g ;

A6: for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p

hence A10: seq is convergent_to_finite_number by A6, MESFUNC5:def 8; :: thesis: ( seq is convergent & lim seq = lim (seq ^\ k) )

hence seq is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)

hence lim seq = lim (seq ^\ k) by A5, A6, A10, MESFUNC5:def 12; :: thesis: verum

( seq is convergent_to_finite_number & seq is convergent & lim seq = lim (seq ^\ k) )

let seq be ExtREAL_sequence; :: thesis: ( seq ^\ k is convergent_to_finite_number implies ( seq is convergent_to_finite_number & seq is convergent & lim seq = lim (seq ^\ k) ) )

set seq0 = seq ^\ k;

assume A1: seq ^\ k is convergent_to_finite_number ; :: thesis: ( seq is convergent_to_finite_number & seq is convergent & lim seq = lim (seq ^\ k) )

then A2: ( not lim (seq ^\ k) = +infty or not seq ^\ k is convergent_to_+infty ) by MESFUNC5:50;

A3: ( not lim (seq ^\ k) = -infty or not seq ^\ k is convergent_to_-infty ) by A1, MESFUNC5:51;

seq ^\ k is convergent by A1, MESFUNC5:def 11;

then A4: ex g being Real st

( lim (seq ^\ k) = g & ( for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

|.(((seq ^\ k) . m) - (lim (seq ^\ k))).| < p ) & seq ^\ k is convergent_to_finite_number ) by A2, A3, MESFUNC5:def 12;

then consider g being Real such that

A5: lim (seq ^\ k) = g ;

A6: for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p

proof

lim (seq ^\ k) = g
by A5;
let p be Real; :: thesis: ( 0 < p implies ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p )

assume 0 < p ; :: thesis: ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p

then consider n being Nat such that

A7: for m being Nat st n <= m holds

|.(((seq ^\ k) . m) - (lim (seq ^\ k))).| < p by A4;

take n1 = n + k; :: thesis: for m being Nat st n1 <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p

end;for m being Nat st n <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p )

assume 0 < p ; :: thesis: ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p

then consider n being Nat such that

A7: for m being Nat st n <= m holds

|.(((seq ^\ k) . m) - (lim (seq ^\ k))).| < p by A4;

take n1 = n + k; :: thesis: for m being Nat st n1 <= m holds

|.((seq . m) - (lim (seq ^\ k))).| < p

hereby :: thesis: verum

let m be Nat; :: thesis: ( n1 <= m implies |.((seq . m) - (lim (seq ^\ k))).| < p )

assume A8: n1 <= m ; :: thesis: |.((seq . m) - (lim (seq ^\ k))).| < p

k <= n + k by NAT_1:11;

then reconsider mk = m - k as Element of NAT by A8, INT_1:5, XXREAL_0:2;

A9: (seq ^\ k) . (m - k) = seq . (mk + k) by NAT_1:def 3;

(n + k) - k <= m - k by A8, XREAL_1:9;

hence |.((seq . m) - (lim (seq ^\ k))).| < p by A7, A9; :: thesis: verum

end;assume A8: n1 <= m ; :: thesis: |.((seq . m) - (lim (seq ^\ k))).| < p

k <= n + k by NAT_1:11;

then reconsider mk = m - k as Element of NAT by A8, INT_1:5, XXREAL_0:2;

A9: (seq ^\ k) . (m - k) = seq . (mk + k) by NAT_1:def 3;

(n + k) - k <= m - k by A8, XREAL_1:9;

hence |.((seq . m) - (lim (seq ^\ k))).| < p by A7, A9; :: thesis: verum

hence A10: seq is convergent_to_finite_number by A6, MESFUNC5:def 8; :: thesis: ( seq is convergent & lim seq = lim (seq ^\ k) )

hence seq is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)

hence lim seq = lim (seq ^\ k) by A5, A6, A10, MESFUNC5:def 12; :: thesis: verum