Exercise 3, § 1

Let E be a metrizable vector space over $\mathbf{R}$, d a translation invariant distance, compatible with the topology of E. Let $|x| = d(x, 0)$ (I, p. 16). Prove that for every integer $n > 0$, we have $\frac{1}{n} |x| \leq \left| \frac{x}{n} \right|$. Deduce from this that if B is a bounded subset of E, then $\sup_{x \in B} |x| < +\infty$ (in other words, B is bounded for the distance d (GT, IX, § 2, No. 3)). Give an example of a metrizable vector space E and an unbounded set in E which is bounded for the distance d (exerc. 2).