Compact Subspace of Metric Space is Bounded

Theorem
Let $M = \left({A, d}\right)$ be a metric space.

Let $C$ be a subspace of $M$.

If $C$ is compact, then it is bounded.

Proof
Let $a \in M$.

Let $N_n \left({a}\right)$ be the $n$-neighborhood of $a$.

Then $\displaystyle C \subseteq \bigcup_{n=1}^\infty N_n \left({a}\right)$ because $\forall x \in C: d \left({x, a}\right) < n$ for some $n \in \N$.

Thus the collection $\left\{{N_n \left({a}\right): n \in \N}\right\}$ forms an open cover of $C$.

Because $C$ is compact, it has a finite subcover, say: $\left\{{N_{n_1} \left({a}\right), N_{n_2} \left({a}\right), \ldots, N_{n_r} \left({a}\right)}\right\}$.

Let $n = \max \left\{{n_1, n_2, \ldots, n_r}\right\}$.

Then $\displaystyle C \subseteq \bigcup_{n=1}^r N_{n_r} \left({a}\right) = N_n \left({a}\right)$.

The result follows by definition of bounded.