normed space of infinite dimension

Question

By $X$ we denote an infinite-dimensional normed space. Show then that exist $x \in X$ and a closed subset $F\subset X$ such that $d(x,F) < \left \| x - y \right \|$ for all $y \in F$.

(Note: It is not required that $F$ is a vector subspace).

Answer 1

Since $X$ is an infinite dimensional normed space, we can use the Riesz lemma to find a sequence $u_n$ such that $\|u_n\|=1$ and $\|u_n-u_m\| > { 1\over 2}$ for all $m\neq n$.

Let $x_n = (1+{1 \over n}) u_n$, and let $F = \{ x_n \}_n$. We see that $\|x_n\| = 1+ {1 \over n} $, and so $d(0,F) = 1 < \|0-y\|$, for any $y \in F$. It remains for us to show that $F$ is closed.

Suppose $n>m \ge 8$, then $\|x_n-x_m\|\ge \|u_n-u_m\|-\| {1 \over n} u_n - {1 \over m} u_m \| > {1 \over 2}-{1 \over 4} = {1 \over 4}$. Hence $\{x_n\}_{n \ge 8}$ consists of isolated points and is closed. Taking the union with $\{x_1,...,x_7\}$ shows that $F$ is closed.