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

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

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

set seq0 = seq ^\ k;

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

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

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

set seq0 = seq ^\ k;

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

per cases
( seq ^\ k is convergent_to_finite_number or seq ^\ k is convergent_to_+infty or seq ^\ k is convergent_to_-infty )
by A1, MESFUNC5:def 11;

end;

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

end;

end;

suppose A2:
seq ^\ k is convergent_to_+infty
; :: thesis: ( seq is convergent & lim seq = lim (seq ^\ k) )

for g being Real st 0 < g holds

ex n being Nat st

for m being Nat st n <= m holds

g <= seq . m

hence A7: seq is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)

lim (seq ^\ k) = +infty by A1, A2, MESFUNC5:def 12;

hence lim seq = lim (seq ^\ k) by A6, A7, MESFUNC5:def 12; :: thesis: verum

end;ex n being Nat st

for m being Nat st n <= m holds

g <= seq . m

proof

then A6:
seq is convergent_to_+infty
by MESFUNC5:def 9;
let g be Real; :: thesis: ( 0 < g implies ex n being Nat st

for m being Nat st n <= m holds

g <= seq . m )

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

for m being Nat st n <= m holds

g <= seq . m

then consider n being Nat such that

A3: for m being Nat st n <= m holds

g <= (seq ^\ k) . m by A2, MESFUNC5:def 9;

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

g <= seq . m

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

g <= seq . m )

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

for m being Nat st n <= m holds

g <= seq . m

then consider n being Nat such that

A3: for m being Nat st n <= m holds

g <= (seq ^\ k) . m by A2, MESFUNC5:def 9;

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

g <= seq . m

hereby :: thesis: verum

let m be Nat; :: thesis: ( n1 <= m implies g <= seq . m )

assume A4: n1 <= m ; :: thesis: g <= seq . m

k <= n + k by NAT_1:11;

then reconsider mk = m - k as Element of NAT by A4, INT_1:5, XXREAL_0:2;

A5: (seq ^\ k) . (m - k) = seq . (mk + k) by NAT_1:def 3;

(n + k) - k <= m - k by A4, XREAL_1:9;

hence g <= seq . m by A3, A5; :: thesis: verum

end;assume A4: n1 <= m ; :: thesis: g <= seq . m

k <= n + k by NAT_1:11;

then reconsider mk = m - k as Element of NAT by A4, INT_1:5, XXREAL_0:2;

A5: (seq ^\ k) . (m - k) = seq . (mk + k) by NAT_1:def 3;

(n + k) - k <= m - k by A4, XREAL_1:9;

hence g <= seq . m by A3, A5; :: thesis: verum

hence A7: seq is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)

lim (seq ^\ k) = +infty by A1, A2, MESFUNC5:def 12;

hence lim seq = lim (seq ^\ k) by A6, A7, MESFUNC5:def 12; :: thesis: verum

suppose A8:
seq ^\ k is convergent_to_-infty
; :: thesis: ( seq is convergent & lim seq = lim (seq ^\ k) )

for g being Real st g < 0 holds

ex n being Nat st

for m being Nat st n <= m holds

seq . m <= g

hence A13: seq is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)

lim (seq ^\ k) = -infty by A1, A8, MESFUNC5:def 12;

hence lim seq = lim (seq ^\ k) by A12, A13, MESFUNC5:def 12; :: thesis: verum

end;ex n being Nat st

for m being Nat st n <= m holds

seq . m <= g

proof

then A12:
seq is convergent_to_-infty
by MESFUNC5:def 10;
let g be Real; :: thesis: ( g < 0 implies ex n being Nat st

for m being Nat st n <= m holds

seq . m <= g )

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

for m being Nat st n <= m holds

seq . m <= g

then consider n being Nat such that

A9: for m being Nat st n <= m holds

(seq ^\ k) . m <= g by A8, MESFUNC5:def 10;

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

seq . m <= g

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

seq . m <= g )

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

for m being Nat st n <= m holds

seq . m <= g

then consider n being Nat such that

A9: for m being Nat st n <= m holds

(seq ^\ k) . m <= g by A8, MESFUNC5:def 10;

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

seq . m <= g

hereby :: thesis: verum

let m be Nat; :: thesis: ( n1 <= m implies seq . m <= g )

assume A10: n1 <= m ; :: thesis: seq . m <= g

k <= n + k by NAT_1:11;

then reconsider mk = m - k as Element of NAT by A10, INT_1:5, XXREAL_0:2;

A11: (seq ^\ k) . (m - k) = seq . (mk + k) by NAT_1:def 3;

(n + k) - k <= m - k by A10, XREAL_1:9;

hence seq . m <= g by A9, A11; :: thesis: verum

end;assume A10: n1 <= m ; :: thesis: seq . m <= g

k <= n + k by NAT_1:11;

then reconsider mk = m - k as Element of NAT by A10, INT_1:5, XXREAL_0:2;

A11: (seq ^\ k) . (m - k) = seq . (mk + k) by NAT_1:def 3;

(n + k) - k <= m - k by A10, XREAL_1:9;

hence seq . m <= g by A9, A11; :: thesis: verum

hence A13: seq is convergent by MESFUNC5:def 11; :: thesis: lim seq = lim (seq ^\ k)

lim (seq ^\ k) = -infty by A1, A8, MESFUNC5:def 12;

hence lim seq = lim (seq ^\ k) by A12, A13, MESFUNC5:def 12; :: thesis: verum