An application of Riesz' Lemma

How does one prove using Riesz' Lemma that an infinite dimensional subspace $Y$ of a Banach space $X$ contains a sequence $\{x_n:n\in \mathbb{N}\}$ in the unit ball of $Y$ such that $n \neq m$ implies that $\|x_n−x_m\|>1/2$?

Solution 1:

Pick any $x_1$ of norm 1. Let $F_1$ be the linear span of $x_1$. Then $F_1$ is finite dimensional and, hence, closed. By Riesz's Lemma, there is an $x_2$ of norm 1 such that $\Vert x_2-\alpha x_1\Vert\ge 1/2$ for all $\alpha$. Let $F_2$ be the linear span of $x_1$ and $x_2$. Then $F_2$ is finite dimensional and, hence, closed. By Riesz's Lemma, there is an $x_3$ of norm 1 such that $\Vert x_3-\alpha x_1-\beta x_3\Vert\ge 1/2$ for all $\alpha$ and all $\beta$. Continue ...