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Math and science::Analysis

Heine-Borel Theorem

Heine-Borel Theorem for $R$

A set $K \subset R$ is compact iff it is closed and bounded.

If you have forgotten some of the formulations of compactness, here is a recap of one:

Compactness

A set $K \subseteq R$ is compact iff every sequence in $K$ has a subsequence that converges to a limit that is also in $K$ .

Heine-Borel Theorem, proof

Below is a copy of Abbott's proof:

Source

Abbott, p84

26.03.2021