Compactness: A set $K \subseteq R$ is compact if every sequence in K has a subsequence that converges to a limit that is also in K
This is the definition of a compact set. I do not understand this definition and how you have to use it to prove a set is compact.
- The main point i do not understand is what "a subsequence" means. And how it converges to a limit.
- The second is, it has to hold for every sequence is there not a randomly weird sequence that just changes a lot. Then i cannot find a limit? Even harder the limit of the subsequence?
So these are multiple questions. Not just one.
A subset is to a set as a subsequence is to a sequence. The only extra requirement that we impose on a subsequence is that its terms are in the same order as in the original sequence. For example, for a sequence $$a_1,a_2,a_3,a_4,a_5,\ldots,$$ a subsequence might be $$a_1,a_4,a_9,a_{16},a_{25},\ldots,$$ but neither $$a_2,a_1,a_3,a_4,a_7,\ldots$$ nor $$a_1,a_1,a_2,a_3,a_5,\ldots$$ would be valid.
This has to hold for every sequence. However, this is somewhat easier that might seem at first glance. For example, by Bolzano-Weierstrass, every bounded sequence in $\mathbb R^n$, no matter how weird it might seem, has a convergent subsequence. I’d recommend you read the proof on the article. That might give some more context on the kinds of things compactness can imply.