show a subset of $\mathbb R^n$ is compact if it is closed and bounded

Question

Use the two lemmas to prove that a subset of $\mathbb R^n$ is compact if it is closed and bounded.

Lemma 1: A closed subset of a compact space is compact

Lemma 2: If $X$ and $Y$ are compact then $X\times Y$ is compact.

I had an idea, but that was quickly shown to be wrong by the answers in this thread

https://math.stackexchange.com/questions/1026318/a-closed-and-bounded-subset-of-rn/1026334#1026334

Thanks

Answer 1

Your idea from the previous thread was almost correct.

HINT: Show that if $A\subseteq\Bbb R^n$ is closed and bounded, then it is a subset of some product of closed intervals (in fact, you can use the same interval!).