[blt]

> Things fall apart when the space is not compact. Your sequences may converge but not to an element in your space.

This is about closedness only. ℝ is not compact but contains its limit points.

[fiforpg]

It could be that the author was thinking about sequential compactness: every sequence of elements of the space has a convergent subsequence (with its limit also in the same space).

For metric spaces, sequential and usual compactness coincide:

https://en.m.wikipedia.org/wiki/Sequentially_compact_space

[Tainnor]

Yes, but not every sequence in R converges.

That's of course also true of a compact space, the author was being imprecise there. What they meant was definitely sequential compactness, as mentioned by a sibling comment.

[Paul-Craft]

Yep. The reason compact spaces are nice is because compactness lets you reduce something that's infinite to something that's finite.