Difference between revisions of "Equivalent statements to compactness of a metric space"

1. Using every sequence in a compact space is a lingering sequence and
2. every lingering sequence has a convergent subsequence

We see that every sequence in a compact space has a convergent subsequence.

Proof of completeness:

To show $(X,d)$ is complete we must show that every Cauchy sequence converges. To do this:

• Let $(x_n)_{n=1}^\infty\subseteq X$ be any Cauchy sequence in $X$
  • Recall that If a subsequence of a Cauchy sequence converges then the Cauchy sequence itself also converges
    • By hypothesis all sequences in $X$ have a convergent subsequence. By this theorem our $(x_n)_{n=1}^\infty$ converges to the same thing.
• As our choice of Cauchy sequence was arbitrary we conclude that all Cauchy sequences converge in $X$, which is the definition of completeness.

Q.E.D.

Proof that $(X,d)$ is totally bounded

TODO: Do this

Q.E.D.