let r be real number ; :: thesis: for seq being Real_Sequence st seq is bounded holds
( r <= lim_sup seq iff for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s )
let seq be Real_Sequence; :: thesis: ( seq is bounded implies ( r <= lim_sup seq iff for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s ) )
set seq1 = superior_realsequence seq;
assume A1: seq is bounded ; :: thesis: ( r <= lim_sup seq iff for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s )
then A2: superior_realsequence seq is bounded by Th58;
thus ( r <= lim_sup seq implies for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s ) :: thesis: ( ( for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s ) implies r <= lim_sup seq )

proof

assume A3: r <= lim_sup seq ; :: thesis: for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s
let s be real number ; :: thesis: ( 0 < s implies for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s )
assume A4: 0 < s ; :: thesis: for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s

proof

let n be Element of NAT ; :: thesis: ex k being Element of NAT st seq . (n + k) > r - s
consider k being Element of NAT such that
A5: seq . (n + k) > ((superior_realsequence seq) . n) - s by A1, A4, Th43;
(superior_realsequence seq) . n >= r by A2, A3, Th10;
then ((superior_realsequence seq) . n) + (seq . (n + k)) > r + (((superior_realsequence seq) . n) - s) by A5, XREAL_1:8;
then seq . (n + k) > ((r + ((superior_realsequence seq) . n)) - s) - ((superior_realsequence seq) . n) by XREAL_1:19;
hence ex k being Element of NAT st seq . (n + k) > r - s ; :: thesis: verum

end;

hence for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s ; :: thesis: verum

end;

assume A6: for s being real number st 0 < s holds
for n being Element of NAT ex k being Element of NAT st seq . (n + k) > r - s ; :: thesis: r <= lim_sup seq
for s being real number st 0 < s holds
lim_sup seq >= r - s

proof

let s be real number ; :: thesis: ( 0 < s implies lim_sup seq >= r - s )
assume A7: 0 < s ; :: thesis: lim_sup seq >= r - s
for n being Element of NAT holds r - s <= (superior_realsequence seq) . n

proof

let n be Element of NAT ; :: thesis: r - s <= (superior_realsequence seq) . n
consider k being Element of NAT such that
A8: r - s < seq . (n + k) by A6, A7;
seq . (n + k) <= (superior_realsequence seq) . n by A1, Th43;
hence r - s <= (superior_realsequence seq) . n by A8, XXREAL_0:2; :: thesis: verum

end;

hence lim_sup seq >= r - s by Th10; :: thesis: verum

end;

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