Every bounded infinite subset of R^k has a limit point

Corollary (Bolzano–Weierstrass, set form): Let $E\subseteq\mathbb{R}^k$ be bounded and infinite. Then $E$ has a limit point ; i.e., there exists $x\in\mathbb{R}^k$ such that every open ball $B(x,\varepsilon)$ contains a point of $E$ different from $x$ (indeed, infinitely many points of $E$).

This is the set-theoretic form of Bolzano–Weierstrass and is a key compactness phenomenon: boundedness plus infinitude forces clustering.

Connection to parent theorem: Choose a sequence of distinct points in $E$. By Bolzano–Weierstrass, it has a convergent subsequence . The subsequential limit is a limit point of $E$.