Given metric space ($X,d$), let {$S_1, S_2,...$} be a set of non-empty sets where $S_1 \supseteq S_2...$, then if $X$ is compact and the $S_t$ are closed then $ \cap_tS_t$ is not empty.
In $\Bbb R$, provide examples that the claim is false if the $S_t$ are closed but not bounded, and the claim is false when $S_t$ are bounded but not closed. I already proved the claim but stuck on the counterexamples. Thanks.
A couple hints - let me know if these are not enough, and I will expand:
(1) What are some closed subsets of $\mathbb{R}$ which are not bounded? (HINT: Think about $\infty$ and $-\infty$ . . .)
(2) What is the intersection of the closed intervals $[0, {1\over n}]$? Do you see how to modify these intervals slightly, so that they are no longer closed and their intersection is the emptyset?