let r be Real; :: thesis: for seq being Real_Sequence st seq is bounded holds
( r <= lim_inf seq iff for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k) )
let seq be Real_Sequence; :: thesis: ( seq is bounded implies ( r <= lim_inf seq iff for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k) ) )
set seq1 = inferior_realsequence seq;
assume A1: seq is bounded ; :: thesis: ( r <= lim_inf seq iff for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k) )
then A2: inferior_realsequence seq is bounded by Th56;
thus ( r <= lim_inf seq implies for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k) ) :: thesis: ( ( for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k) ) implies r <= lim_inf seq )

proof

assume A3: r <= lim_inf seq ; :: thesis: for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k)
let s be Real; :: thesis: ( 0 < s implies ex n being Nat st
for k being Nat holds r - s < seq . (n + k) )
assume A4: 0 < s ; :: thesis: ex n being Nat st
for k being Nat holds r - s < seq . (n + k)
ex n being Nat st
for k being Nat holds r - s < seq . (n + k)

proof

consider n1 being Nat such that
A5: (inferior_realsequence seq) . n1 > (upper_bound (inferior_realsequence seq)) - s by A2, A4, Th7;
((inferior_realsequence seq) . n1) + (upper_bound (inferior_realsequence seq)) > r + ((upper_bound (inferior_realsequence seq)) - s) by A3, A5, XREAL_1:8;
then ((inferior_realsequence seq) . n1) + (upper_bound (inferior_realsequence seq)) > (r - s) + (upper_bound (inferior_realsequence seq)) ;
then A6: (((inferior_realsequence seq) . n1) + (upper_bound (inferior_realsequence seq))) - (upper_bound (inferior_realsequence seq)) > r - s by XREAL_1:20;

now :: thesis: for k being Nat holds r - s < seq . (n1 + k)

let k be Nat; :: thesis: r - s < seq . (n1 + k)
(inferior_realsequence seq) . n1 <= seq . (n1 + k) by A1, Th40;
then (r - s) + ((inferior_realsequence seq) . n1) < (seq . (n1 + k)) + ((inferior_realsequence seq) . n1) by A6, XREAL_1:8;
then ((r - s) + ((inferior_realsequence seq) . n1)) - ((inferior_realsequence seq) . n1) < seq . (n1 + k) by XREAL_1:19;
hence r - s < seq . (n1 + k) ; :: thesis: verum

end;

hence ex n being Nat st
for k being Nat holds r - s < seq . (n + k) ; :: thesis: verum

end;

hence ex n being Nat st
for k being Nat holds r - s < seq . (n + k) ; :: thesis: verum

end;

assume A7: for s being Real st 0 < s holds
ex n being Nat st
for k being Nat holds r - s < seq . (n + k) ; :: thesis: r <= lim_inf seq
for s being Real st 0 < s holds
r - s <= lim_inf seq

proof

let s be Real; :: thesis: ( 0 < s implies r - s <= lim_inf seq )
assume 0 < s ; :: thesis: r - s <= lim_inf seq
then consider n1 being Nat such that
A8: for k being Nat holds r - s < seq . (n1 + k) by A7;
for k being Nat holds r - s <= seq . (n1 + k) by A8;
then A9: r - s <= (inferior_realsequence seq) . n1 by A1, Th42;
(inferior_realsequence seq) . n1 <= upper_bound (inferior_realsequence seq) by A2, Th7;
hence r - s <= lim_inf seq by A9, XXREAL_0:2; :: thesis: verum

end;

hence r <= lim_inf seq by XREAL_1:57; :: thesis: verum