The Bolzano-Weierstrass and Heine-Borel Theorem

Question

Why do the Bolzano-Weierstrass, Heine-Borel theorem and the general notion of compactness matter? I understand that they are used in the result that power series are continuous, but this alone does not seem to give a general reason to why they are so forced in elementary mathematics. What caused for these ideas to be abstracted?

The connection with completeness, in the sense that Cauchy sequences converge, and that a compact space is, in a sense, invariant to how it is placed into another topological space may be relevant to an explanation.

Answer 1

Here are a few reasons:

• Every continuous from a compact set into $\mathbb R$ has a maximum and a minimum. In particular, they are bounded.
• The Bolzano-Wierstrass theorem is used to prove that $\mathbb R$ is complete. It follows from this that, in $\mathbb R$, every absolutely convergente series converges. And it is easier to study series of non-negative terms than series in general.
• As a consequence of the fact that $\mathbb R$ is complete, it becomes easy to define new functions, such as$$x\mapsto\sum_{n=1}^\infty\frac{x^n}{n^n}.$$
• On a compact set, every continuous function is uniformly continuous. This leads to a very easy proof of the fact that every continuous function from an interval $[a,b]$ into $\mathbb R$ is Riemann-integrable.