It seems very intuitive and simple, but how would I go about proving something like this? Thanks.
Yes, this is true, and it follows from Weirstrass' Thm : every bounded , infinite subset has a limit point. And this generalizes to $\mathbb R^n$
More formally, cover $\mathbb R^n$ by balls $B(0,n) ; n=1,2,3,... $ , i.e., for all
$n $ in $\mathbb N$ Now, by a straight-forward cardinality argument (countable union of countable is countable) , there will be at least one ball $B(0, n_k)$containing infinitely-many points. Then apply Weirstrass' theorem.