Let $a_n$ be a bounded sequence and consider the following sequence. $b_n:=\text{sup}\{(a_n,a_{n+1},a_{n+2}\cdots\})$ Prove that $b_n$ converges:
My proof: $b_n$ is monotonically decreasing by the fact that for the set in each $b_n$, $b_{n+1}$ can only have maximums removed, not added. This means that the supremum of the set can only stay the same or decrease, meaning the sequence is monotonically decreasing.
$b_n$ is bound below by $\text{inf}(\{a_n:n\in\mathbb{N}\})$, $\forall n \ a_n>\text{inf}(\{a_n:n\in\mathbb{N}\})\implies$ that all upper bounds of any subsequence must be greater than $\text{inf}(\{a_n:n\in\mathbb{N}\})$, which implies that $\text{sup}\{(a_n,a_{n+1},a_{n+2}\cdots\})$ must be greater than $\text{inf}(\{a_n:n\in\mathbb{N}\})$.
This sequence is both bounded below and monotonically decreasing, meaning it is convergent.
let $b_n = \sup_n \{a_n,....\}$
since the sequence $a_n$ is bounded hence $b_n$ is bounded below with $\inf_n\{a_1,....\}$
also it's an easy verification that $b_n$ is a decreasing sequence, hence it's convergent.