Question

For
[infinity]
N₀ ∊ N, let {aₙ}ₙ₌₁
be a bounded sequence such that
aₙ₊₁≤=aₙ, for all n>N₀. Prove that
[infinity]
{aₙ}ₙ₌₁ converges to i=inf(R) where R={aₙ : n > N₀, n ∊ N} is a subset of im(aₙ)