let n be Nat; :: thesis: for seq being Real_Sequence st seq is increasing & rng seq c= NAT holds
n <= seq . n
let seq be Real_Sequence; :: thesis: ( seq is increasing & rng seq c= NAT implies n <= seq . n )
defpred S₁[ Nat] means $1 <= seq . $1;
assume that
A1: seq is increasing and
A2: rng seq c= NAT ; :: thesis: n <= seq . n
A3: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A4: k <= seq . k ; :: thesis: S₁[k + 1]
k + 1 in NAT ;
then k + 1 in dom seq by FUNCT_2:def 1;
then seq . (k + 1) in rng seq by FUNCT_1:def 3;
then reconsider k9 = seq . (k + 1) as Element of NAT by A2;
seq . k < seq . (k + 1) by A1, SEQM_3:def 6;
then k < k9 by A4, XXREAL_0:2;
hence S₁[k + 1] by NAT_1:13; :: thesis: verum

end;

0 in NAT ;
then 0 in dom seq by FUNCT_2:def 1;
then seq . 0 in rng seq by FUNCT_1:def 3;
then A5: S₁[ 0 ] by A2;
for n being Nat holds S₁[n] from NAT_1:sch 2(A5, A3);
hence n <= seq . n ; :: thesis: verum