let n be Element of NAT ; :: thesis: for seq being Real_Sequence st seq is non-increasing & seq is bounded_below holds
( (inferior_realsequence seq) . n = inf seq & inferior_realsequence seq is V8() )
let seq be Real_Sequence; :: thesis: ( seq is non-increasing & seq is bounded_below implies ( (inferior_realsequence seq) . n = inf seq & inferior_realsequence seq is V8() ) )
assume A1: ( seq is non-increasing & seq is bounded_below ) ; :: thesis: ( (inferior_realsequence seq) . n = inf seq & inferior_realsequence seq is V8() )
defpred S₁[ Nat] means (inferior_realsequence seq) . $1 = inf seq;
for n being Nat holds (inferior_realsequence seq) . n = inf seq

proof

A2: S₁[ 0 ] by A1, Th40;
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
k in NAT by ORDINAL1:def 13;
hence ( S₁[k] implies S₁[k + 1] ) by A1, Th74; :: thesis: verum

end;

thus for k being Nat holds S₁[k] from NAT_1:sch 2(A2, A3); :: thesis: verum

end;

hence ( (inferior_realsequence seq) . n = inf seq & inferior_realsequence seq is V8() ) by VALUED_0:def 18; :: thesis: verum