Let $A$ be a non-empty compact set in a normed space $E$. Prove that there exists a finite or countable subset of $A$ which is dense in $A$.

Question

Let $A$ be a non-empty compact set in a normed space $E$. Prove that there exists a finite or countable subset of $A$ which is dense in $A$.

If $A$ is finite or countable, it is easy using the fact that that a countable union of countable sets is countable ; we consider all points of $A$ and sequences of $A$ that converge towards these points.

Do you have a hint for the general case?

Answer 1

For every integer $n>0$, consider the open covering $B(x,1/n), x\in A$, it has a finite subcovering $B(x^n_1,1/n),...,B(x^n_{n_p},1/n)$. The set $x^i_j$ is dense.

Consider a Ball $B(x,r)$, and $1/n< r$, cover $A$ with $B(x^n_i,1/n)$, $x\in B(x^n,1/n)$. This implies that $d(x,x_i)<1/n<r$.