Some concerns about using "a subset $X$ of a metric space is closed if and only if every convergent sequence in $X$ has their limit in $X$ "

I've seen this proof and I have some concerns about using this theorem

In some questions, we know the given set $X$ is closed

Based on the information given that $X$ is closed, is that always admissible to claim " considering an arbitrary convergent sequence $(x_n)\subset X$ with its limit $x^*\in X$"

I am wobbling about this claim because the theorem says every convergent sequence converges within the space. Given an arbitrary $x^*\in X$, what if there is no sequence in $X$ converges to $x^*$