Compact subsets of $\Bbb R$ and $\Bbb R^k$

Question

I'm confused regarding the following ideas:

1. I understand that if a set $A$ is compact then it is bounded and closed. This works in both cases if $A$ is in $\Bbb R$ or in $\Bbb R^k$. But is the opposite true? If a set is bounded and closed in $\Bbb R$ is it compact? If a set is bounded and closed in $\Bbb R^k$, is it compact?
2. What does it mean to say that the compact subsets of $\Bbb R$ are the closed and bounded subsets of $\Bbb R$, and similarly for $\Bbb R^k$?

Answer 1

Yes, if a subset of $\mathbb{R}$ (or $\mathbb{R}^k$) is closed and bounded, then it is compact (though this is a nontrivial theorem).

"The compact subsets of $\mathbb{R}$ are the closed and bounded subsets of $\mathbb{R}$" means that if $A\subseteq\mathbb{R}$, then $A$ is compact iff $A$ is closed and bounded. That is, compact subsets of $\mathbb{R}$ are closed and bounded and closed and bounded subsets of $\mathbb{R}$ are compact.

Answer 2

Actually, since the implication works two ways, when dealing solely in $\mathbb R$ or $\mathbb R^n$, many textbooks and courses take closed and bounded as THE definition of compactness (both Folland and Edward's Jr do); and never even mention the open cover definition. Other's do, like Lay. I've seen it done both ways without any real loss or gain of material to study in the course. It turns out that closed and bounded is frequently a more useful definition of compactness in $\mathbb R$, as well.

Make sure you remember that this theorem is an if and only if: a subset S of $\mathbb R^n$ is compact if and only if it is closed and bounded.