Bolzano-Weierstraß Theorem

Every Bounded infinite set in $\Bbb{R}^n$ has an Accumulation Point. For $n=1$, the theorem can be stated as follows: If a Set in a Metric Space, finite-dimensional Euclidean Space, or First-Countable Space has infinitely many members within a finite interval $x\in [a,b]$, then it has at least one Limit Point $x$ such that $x\in [a,b]$. The theorem can be used to prove the Intermediate Value Theorem.