If a set is closed and unbounded, is it still possible for it to be sequentially compact?

Question

Sorry if this is a trivial question, but I couldn't find an answer for it yet. I know a set $S$ is compact iff every open cover of $S$ has a finite subcover. I also see how this is not the case for the closed and unbounded set $A = [1,\infty)$. But isn't $A$ still sequentially compact? if $x_n \rightarrow \infty$, it doesn't converge, and $x_n \rightarrow a < \infty \Rightarrow a \in S$. And if we're in a normed vector space, wouldn't this be a set that is sequentially compact but has no finite subcover?