Norm on Vector Space is Continuous Function

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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 $V$.

Then we have:

$\norm {x_n - x} \to 0$

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.

$\blacksquare$