Suppose we have the sequence $a_n$ such that $a_n \leq \sup\{a_k : k \geq n\}$ for all $n \in \mathbb{N}$ and $\lim_{n\rightarrow\infty}\space a_n$ exists.
Can we conclude that $\lim_{n\rightarrow\infty}\space a_n \leq \lim_{n\rightarrow\infty}\space \sup\{a_k : k \geq n\}$?
It seems like this is the case and I would like to use it in one of my proofs, but I can't figure out if it is true or not.
Is there a proof whether this is true or not, and if it is not true in general, what are the circumstances where it is true?
If the sequence $a_n$ converges, then
$$\lim_{n\to\infty} a_n = \lim_{n\to\infty} \sup\{a_k| k\geq n\}$$
which is even stronger (equality rather than inequality) than what you are asking.