Normed Vector Space is Open in Itself/Proof 1

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

Then the set $X$ is an open set of $M$.

Proof
By definition, an open set $S \subseteq A$ is one where every point inside it is an element of an open ball contained entirely within that set.

Let $x \in X$.

An open ball of $x$ in $M$ is by definition a subset of $X$.

Hence the result.