Norm on Vector Space is Continuous Function

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Theorem

Let $V$ be a vector space with norm $\|\cdot\|$.

The function $\|\cdot\| : V \to \R$ is continuous.


Proof

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

$x_n \to x \implies \|x_n - x\| \to 0$.

By the Reverse Triangle Inequality $|\|x_n\| - \|x\|| \leq \|x_n - x\|$.

Hence, $|\|x_n\| - \|x_n\|| \to 0$.

Thus $\|x_n\| \to \|x\|$

$\blacksquare$