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# Class Contents

## Compactness and Sequential Compactness are Equivalent (continued)

### On with the proof of Theorem 6.24

Our standing assumption throughout the remainder of this section will be
that `X` is a metric space and that `K` is a sequentially compact
subset of `X`. Our aim will be to show that `K` is
compact. To this end, suppose further that we have been presented with a
fixed open cover `U` of `K`. Our job will be to show that
we can pick out a finite cover of `K` consisting of points of `U`.

**Claim:** We can use Lemma 6.25 to find a sequence `(x _{n})` of
points in

Remark:

We call the set of points `(x _{n})` a

Now we use the `x _{n}` to select out a collection of
open sets from the open cover which we can work with.

**Claim:** The sets `U _{{m,n}}` cover

The picture shows the details of this clearly:

Question:
Work out the details of why `B _{{m,n}}` is contained in

We are now ready for the main part of the argument. We shall show that
finitely many of the `U _{{m,n}}` will cover

Factoid:

Factoid:

The next class, which is the last one of this chapter, will introduce a new property of metric spaces: completeness. This allows us to answer the question "When does a sequence converge?", without knowing in advance what it is supposed to converge to.