For example, [1,3] is a closed and bounded subset of metric space $\mathbb{R}$, so the set should be compact. But consider the open cover $(0.9,1.1)\cup(2.9, 3.1)\cup(2-\frac{1}{n}, 2+\frac{1}{n})$, where $n \in \mathbb{Z^+} $is from 1 to $\infty$, this open cover is infinite. So this set should not be compact.
What's wrong?
Thanks.