let seq be ExtREAL_sequence; :: thesis: for rseq being Real_Sequence st seq = rseq & seq is convergent_to_finite_number holds
( rseq is convergent & lim seq = lim rseq )
let rseq be Real_Sequence; :: thesis: ( seq = rseq & seq is convergent_to_finite_number implies ( rseq is convergent & lim seq = lim rseq ) )
assume that
A1: seq = rseq and
A2: seq is convergent_to_finite_number ; :: thesis: ( rseq is convergent & lim seq = lim rseq )
A3: ( not lim seq = +infty or not seq is convergent_to_+infty ) by A2, MESFUNC5:50;
A4: ( not lim seq = -infty or not seq is convergent_to_-infty ) by A2, MESFUNC5:51;
seq is convergent by A2, MESFUNC5:def 11;
then consider g being real number such that
A5: lim seq = g and
A6: 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 and
seq is convergent_to_finite_number by A3, A4, MESFUNC5:def 12;
A7: for p being real number st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((rseq . m) - g) < p

proof

let p be real number ; :: thesis: ( 0 < p implies ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((rseq . m) - g) < p )
assume 0 < p ; :: thesis: ex n being Element of NAT st
for m being Element of NAT st n <= m holds
abs ((rseq . m) - g) < p
then consider n being Nat such that
A8: for m being Nat st n <= m holds
|.((seq . m) - (lim seq)).| < p by A6;
reconsider N = n as Element of NAT by ORDINAL1:def 12;
take N ; :: thesis: for m being Element of NAT st N <= m holds
abs ((rseq . m) - g) < p

hereby :: thesis: verum

let m be Element of NAT ; :: thesis: ( N <= m implies abs ((rseq . m) - g) < p )
assume N <= m ; :: thesis: abs ((rseq . m) - g) < p
then A9: |.((seq . m) - (lim seq)).| < p by A8;
g is Real by XREAL_0:def 1;
then (rseq . m) - g = (seq . m) - (lim seq) by A1, A5, SUPINF_2:3;
hence abs ((rseq . m) - g) < p by A9, EXTREAL2:1; :: thesis: verum

end;

end;

then rseq is convergent by SEQ_2:def 6;
hence ( rseq is convergent & lim seq = lim rseq ) by A5, A7, SEQ_2:def 7; :: thesis: verum