Compact Set (Compactness)

compact set, compactness

Let $K \subseteq R$ . $K$ is compact if every Sequence of points in $K$ has a subsequence which converges to a point $k \in K$ .

Some examples include:

Let $a < b$ . Then $[a, b]$ is compact. Why? If I have a sequence of points $(x_{n}) \in [a, b]$ for all $n \in N$ then $\forall n \in N$ then $a \leq x_{n} \leq b$ . This means the sequence is bounded. So then apply the Balzano-Weierstrass Theorem to get that there $\exists$ a convergent subsequence $(x_{n_{k}})$ that converges.

But for each $k \in N$ then $a \leq x_{n_{k}} \leq b$ still. We know $(x_{n_{k}})$ is convergent, so say $(x_{n_{k}}) \to x$ so by the Limits and Order (Order Limit Theorem) then $a \leq x \leq b$ . We know $x \in [a, b]$ as a result.
As a counter example $R$ is not compact. The sequence of numbers:

(1, 2, 3, 4, \dots) \subseteq R

has no convergent subsequence.