Finite Subset of Normed Vector Space is Closed

Theorem
Let $M = \struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

Let $F \subseteq X$ be finite.

Then $F$ is closed in $M$.

Proof
Suppose, $F$ is empty.

By Empty Set is Closed in Normed Vector Space, $F$ is closed.

Suppose, for some $n \in \N$:


 * $\displaystyle F = \bigcup_{i \mathop = 1}^n \set {x_i}$

We have that Singleton in Normed Vector Space is Closed.

Hence, $F$ is a finite union of closed sets.

By Finite Union of Closed Sets is Closed in Normed Vector Space, $F$ is closed.