Supremum of a sequence - same as supremum of set? This is more of a notational/terminology question than anything. I was refreshing my knowledge on the lemmas in the monotone convergence theorem, and saw the claim "If a sequence of real numbers is increasing and bounded above, then its supremum is the limit." I haven't really thought of sequences having a supremum, as I thought this was more reserved for sets. So if $a_n$ is a sequence defined in $\mathbb{N}$, then are we defining the supremum of $a_n$ as $\sup(\\{a_n | n \in \mathbb{N}\\})$? If not, how do we extend the notion of supremum to sequences?

Yes, the supremum of a sequence $(a_n)_{n\in\mathbb N}$ is $\sup\\{a_n\,|\,n\in\mathbb{N}\\}$. More generally, the supremum of a real function with doain $X$ is $\sup\\{f(x)\,|\,x\in X\\}$.