Norm on Vector Space is Continuous Function

Theorem
Let $V$ be a vector space with norm $\norm {\, \cdot \,}$.

The function $\norm {\, \cdot \,}: V \to \R$ is continuous.

Proof
Let $x_n \to x$ in $V$.

We have:
 * $x_n \to x \implies \norm {x_n - x} \to 0$

By the Reverse Triangle Inequality:
 * $\size {\norm {x_n} - \norm x} \le \norm {x_n - x}$

Hence:
 * $\size {\norm {x_n} - \norm x} \to 0$

Thus:
 * $\norm {x_n} \to \norm x$

Hence the result from the definition of a continuous real function.