Space of Bounded Sequences with Supremum Norm forms Normed Vector Space

Theorem
Space of bounded sequences with supremum norm forms a normed vector space.

Proof
We have that:


 * Space of bounded sequences is a vector space


 * Supremum norm on the space of bounded sequences is a norm

By definition, $\struct {\ell^\infty, \norm {\, \cdot \,}_\infty}$ is a normed vector space.