let seq be ExtREAL_sequence; :: thesis: for rseq being Real_Sequence st seq = rseq holds

( seq is bounded_above iff rseq is V70() )

let rseq be Real_Sequence; :: thesis: ( seq = rseq implies ( seq is bounded_above iff rseq is V70() ) )

assume A1: seq = rseq ; :: thesis: ( seq is bounded_above iff rseq is V70() )

then consider q being Real such that

A4: for n being Nat holds rseq . n < q by SEQ_2:def 3;

hence rng seq is bounded_above by XXREAL_2:def 10; :: according to RINFSUP2:def 4 :: thesis: verum

( seq is bounded_above iff rseq is V70() )

let rseq be Real_Sequence; :: thesis: ( seq = rseq implies ( seq is bounded_above iff rseq is V70() ) )

assume A1: seq = rseq ; :: thesis: ( seq is bounded_above iff rseq is V70() )

hereby :: thesis: ( rseq is V70() implies seq is bounded_above )

assume
rseq is V70()
; :: thesis: seq is bounded_above assume
seq is bounded_above
; :: thesis: rseq is V70()

then rng rseq is bounded_above by A1;

then consider p being Real such that

A2: p is UpperBound of rng rseq by XXREAL_2:def 10;

A3: for y being Real st y in rng rseq holds

y <= p by A2, XXREAL_2:def 1;

end;then rng rseq is bounded_above by A1;

then consider p being Real such that

A2: p is UpperBound of rng rseq by XXREAL_2:def 10;

A3: for y being Real st y in rng rseq holds

y <= p by A2, XXREAL_2:def 1;

now :: thesis: for n being Nat holds rseq . n < p + 1

hence
rseq is V70()
by SEQ_2:def 3; :: thesis: verumlet n be Nat; :: thesis: rseq . n < p + 1

n in NAT by ORDINAL1:def 12;

then rseq . n in rng rseq by FUNCT_2:4;

then 0 + (rseq . n) < 1 + p by A3, XREAL_1:8;

hence rseq . n < p + 1 ; :: thesis: verum

end;n in NAT by ORDINAL1:def 12;

then rseq . n in rng rseq by FUNCT_2:4;

then 0 + (rseq . n) < 1 + p by A3, XREAL_1:8;

hence rseq . n < p + 1 ; :: thesis: verum

then consider q being Real such that

A4: for n being Nat holds rseq . n < q by SEQ_2:def 3;

now :: thesis: for r being ExtReal st r in rng seq holds

r <= q

then
q is UpperBound of rng seq
by XXREAL_2:def 1;r <= q

let r be ExtReal; :: thesis: ( r in rng seq implies r <= q )

assume r in rng seq ; :: thesis: r <= q

then ex x being object st

( x in dom rseq & r = rseq . x ) by A1, FUNCT_1:def 3;

hence r <= q by A4; :: thesis: verum

end;assume r in rng seq ; :: thesis: r <= q

then ex x being object st

( x in dom rseq & r = rseq . x ) by A1, FUNCT_1:def 3;

hence r <= q by A4; :: thesis: verum

hence rng seq is bounded_above by XXREAL_2:def 10; :: according to RINFSUP2:def 4 :: thesis: verum