let seq be ExtREAL_sequence; :: thesis: for rseq being Real_Sequence st seq = rseq holds
( seq is bounded_below iff rseq is V71() )
let rseq be Real_Sequence; :: thesis: ( seq = rseq implies ( seq is bounded_below iff rseq is V71() ) )
assume A1: seq = rseq ; :: thesis: ( seq is bounded_below iff rseq is V71() )
hereby :: thesis: ( rseq is V71() implies seq is bounded_below )
assume seq is bounded_below ; :: thesis: rseq is V71()
then rng rseq is bounded_below by A1;
then consider p being Real such that
A2: p is LowerBound of rng rseq by XXREAL_2:def 9;
A3: for y being Real st y in rng rseq holds
p <= y by A2, XXREAL_2:def 2;
now :: thesis: for n being Nat holds p - 1 < rseq . n
let n be Nat; :: thesis: p - 1 < rseq . n
n in NAT by ORDINAL1:def 12;
then rseq . n in rng rseq by FUNCT_2:4;
then p - 1 < (rseq . n) - 0 by A3, XREAL_1:15;
hence p - 1 < rseq . n ; :: thesis: verum
end;
hence rseq is V71() by SEQ_2:def 4; :: thesis: verum
end;
assume rseq is V71() ; :: thesis: seq is bounded_below
then consider q being Real such that
A4: for n being Nat holds q < rseq . n by SEQ_2:def 4;
now :: thesis: for r being ExtReal st r in rng seq holds
q <= r
let r be ExtReal; :: thesis: ( r in rng seq implies q <= r )
assume r in rng seq ; :: thesis: q <= r
then ex x being object st
( x in dom rseq & r = rseq . x ) by A1, FUNCT_1:def 3;
hence q <= r by A4; :: thesis: verum
end;
then q is LowerBound of rng seq by XXREAL_2:def 2;
hence rng seq is bounded_below by XXREAL_2:def 9; :: according to RINFSUP2:def 3 :: thesis: verum