real analysis - show that if every sequence of $K$ has a subsequence converging to a point in $K$, show that $K$ is closed and bounded

I am able to prove the closed part of the proof but I'm having trouble proving the bounded part.

We can use proof by contradiction and say suppose $K$ is not bounded.
Then $x_n \to \infty$, and there is a subsequence $x_{n_k} \to x$ in $K$ by assumption. But an unbounded sequence can still have a converging subsequence. I'm not sure how to go from here.