Proving that every norm is continuous with respect to $\lVert\cdot\rVert_\infty$

Question

Let $N$ be a norm on a finite dimensional $\mathbb R$-vector space. I'd like to prove that $N$ is continuous with respect to the max-norm $\lVert\cdot\rVert_\infty$.

1. I can prove continuity at zero by using a simple argument to show $N$ is bounded on the $\lVert\cdot\rVert_\infty$ unit sphere.
2. After that, $\lim_{\epsilon\to 0} N(x+\epsilon)\leq N(x)$ follows by sublinearity.

But how can I get $\lim_{\epsilon\to 0} N(x+\epsilon)\geq N(x)$?

Answer 1

For any $\epsilon$ in the space, $$N(x)=N(x+\epsilon-\epsilon)\le N(x+\epsilon)+N(\epsilon)$$