Supremum Norm is Norm

Theorem
Let $S$ be a set.

Let $\left({X, \left\Vert{\, \cdot \,}\right\Vert}\right)$ be a normed vector space over $K \in \left\{{\R, \C}\right\}$.

Let $\mathcal B$ be the set of bounded mappings $S \to X$.

Let $\left\Vert{\, \cdot \,}\right\Vert_\infty$ be the supremum norm on $\mathcal B$.

Then $\left\Vert{\, \cdot \,}\right\Vert_\infty$ is a norm on $\mathcal B$.

Proof
First:

Now let $\lambda \in K, f \in \mathcal B$

We have:

Finally let $f, g \in \mathcal B$.

We have:

Thus $\left\Vert{\cdot}\right\Vert_\infty$ has the defining properties of a norm.