Are Compact Intervals also Compact Sets? I know that "compact intervals" are closed and bounded intervals and that "compact sets" are sets with every infinite sequence of points having an infinite subsequence that converges to some point of the set
But are Compact Intervals also Compact Sets?
 A: Yes! A compact interval is just a connected compact set. In $\mathbb R$ the two characterizations for compactness are the same. This it the Heine Borel theorem, which states that every compact set is closed and bounded. 
For clarity's sake there are three definitions of compactness that are worth mentioning here, and all are (nontrivially) equivalent for $\mathbb R$:
Sequential Compactness: A space $X$ is said to be sequentially compact if every infinite sequence $(a_n) \subseteq X$ has a convergent subsequence.
Compactness: A  space $X$ is said to be compact if every open cover admits a finite subcover. In other words, if $X  \subseteq \bigcup_{\alpha \in A} U_{\alpha}$ where each $U_{\alpha}$ is open, then there exists a finite collection from this family so that $$X \subseteq \bigcup_{i=1}^{N} U_{i}.$$
Closed and Bounded: this is standard.
A: You might also look at the theorem of Bernard Bolzano/Karl Weierstraß that every bounded sequence in $\Bbb R^n$ has a convergent sub-sequence, which is the same as to say that all bounded sets in $\Bbb R^n$ are (sequentially) pre-compact.
