Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?

Although they coincide in metric spaces, in general sequential compactness and compactness are very different properties: neither implies the other, so you can’t expect to use sequential compactness to prove something about compactness. Thus, you’re really talking about two different theorems.

Theorem 1. If $\langle X,\tau\rangle$ is a compact space, and $K$ is a closed subset of $X$, then $K$ is compact.

The proof using open covers is trivial.

Theorem 2. If $\langle X,\tau\rangle$ is a sequentially compact space, and $K$ is a closed subset of $X$, then $K$ is sequentially compact.

Proof. Let $\sigma=\langle x_k:k\in\Bbb N\rangle$ be a sequence in $K$. Since $\sigma$ is also a sequence in $X$, it has a subsequence $\langle x_{n_k}:k\in\Bbb N\rangle$ that converges to some $x\in X$. If $U$ is an open neighborhood of $x$, there is a $k\in\Bbb N$ such that $x_{n_k}\in U$, so $U\cap K\ne\varnothing$. Thus, $x\in\operatorname{cl}K=K$, and the subsequence $\langle x_{n_k}:k\in\Bbb N\rangle$ is convergent in $K$ as well as in $X$. $\dashv$

As you can see, it’s not hard to prove that sequential compactness is inherited by closed sets, but the proof is at least a little more involved than the trivial proof for compactness.

Since we're dealing with subsets of a metric space, sequential compactness and compactness are the same thing, so we can instead show that a closed subset of a sequentially compact set is also sequentially compact. To see this, suppose $X$ is sequentially compact and $Y$ a closed subset of $X$. Take any sequence $\{x_n\}$ of points of $Y$. This is a sequence of points of $X$, so what can we say about it based on the sequential compactness of $X$? Since $Y$ is closed (contains all its limit points), what can we then conclude?

In general, "sequentially compact" and "compact" are not the same thing. (Neither of them even need be a consequence of the other!) In such settings, you're going to need to show that open covers have finite subcovers, or show that non-empty collections of closed sets with the finite intersection property have non-empty intersections, or something like that.