Compactness/Heine-Borel/Fact

The Theorem of Heine-Borel

Let ${}T\subseteq \mathbb {R} ^{n}$ be a subset. Then the following statements are equivalent.

${}T$ is compact.
Every sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ in ${}T$ has an accumulation point in ${}T$ .
Every sequence ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ in ${}T$ has a subsequence that converges in ${}T$ .
${}T$ is closed and bounded.