Show that every infinite dimensional normed linear space contains a sequence $(x_n)$ such that $\|x_n\|=1$ $\forall$ and $\|x_m-x_n\| \geq 1$ for all $m ,n \in \mathbb{N}$ and $m \neq n$.
I tried to show it by using the fact that every infinite dimensional norm linear space admits a linear discontinuous functional. But I am stuck and the idea is leading nowhere. Also if that was something less than 1 we could use Riesz's Lemma but here it doesn't seem to work. I would really appreciate some help. Thanks in advance!!
