Prove that point is not in set.

Question

Suppose $x \in \mathbb{R}^n$ and $\|x\| > 1$ with $\|x\| := \inf\left\{ \lambda > 0 \mid x/\lambda\in B \right\} $ where B is open, convex, symmetric and bounded.

How can we show that $x \notin B$?

I see that we would need $ \lambda = 1 $ such that $x/\lambda = x \in B$.

I feel a bit dumb, thank you for your effort!

Answer 1

If $x\in B$ then $x=\frac x {\lambda}\in B$ with $\lambda = 1$, thus $\mathrm {inf}\left\{\lambda\gt0|\frac x \lambda \in B\right\}\le 1$ from the definition of $\mathrm {inf}$, which is against the hypothesis $||x||\gt1$