let seq be ExtREAL_sequence; :: thesis: inferior_realsequence seq is non-decreasing

now

let n, m be Element of NAT ; :: thesis: ( m <= n implies (inferior_realsequence seq) . m <= (inferior_realsequence seq) . n )
consider Y being non empty Subset of ExtREAL such that
A1: Y = { (seq . k) where k is Element of NAT : m <= k } and
A2: (inferior_realsequence seq) . m = inf Y by Def6;
consider Z being non empty Subset of ExtREAL such that
A3: Z = { (seq . k) where k is Element of NAT : n <= k } and
A4: (inferior_realsequence seq) . n = inf Z by Def6;
assume A5: m <= n ; :: thesis: (inferior_realsequence seq) . m <= (inferior_realsequence seq) . n
Z c= Y

proof

let z be set ; :: according to TARSKI:def 3 :: thesis: ( not z in Z or z in Y )
assume z in Z ; :: thesis: z in Y
then consider k being Element of NAT such that
A6: z = seq . k and
A7: n <= k by A3;
m <= k by A5, A7, XXREAL_0:2;
hence z in Y by A1, A6; :: thesis: verum

end;

hence (inferior_realsequence seq) . m <= (inferior_realsequence seq) . n by A2, A4, XXREAL_2:60; :: thesis: verum

end;

hence inferior_realsequence seq is non-decreasing by Th7; :: thesis: verum