A metric space $X$ is said to be discrete if every point is isolated.
A point $x ∈ A ⊂ X$ is an isolated point of $A$ if some open ball centred at $x$ contains no members of $A$ other than $x$ itself.
I am having troubles with proving the following statement:
Every infinite metric space $(X, d)$ contains an infinite subset $A$ such that $(A, d)$ is discrete.
I have spent some time on this problem. I am thinking that a constructive proof may be impossible. But even if I tried proof by contradiction, I still did not get much progress. Can someone help me? Thanks so much.