Supremum of a sequence - same as supremum of set?

Question

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?

Answer 1

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 $\sup\{f(x)\,|\,x\in X\}$.

Answer 2

The supremum is the supremum of the range of the sequence $(a_{n})$, that is, $\sup\{a_{n}: n=1,2,...\}$.

So several sequences may have the same supremum, especially when they have the same range:

$(x_{n})=(1,0,1,0,1,0,...)$ and $(y_{n})=(0,1,0,1,0,1,...)$ are different sequences, but they have the same supremum $1$.