As it is also asked in here, in an example in the book Analysis of Manifold by Munkres,
It is given a subspace of $\mathbb{R}^n$ that is closed and bounded, but not compact.
However, in the book it is stated that
Theorem 4.9: If $X$ is a closed and bounded subset of $\mathbb{R}^n $, then $X$ is compact.
So why does the example and the theorem not contradict with each other ?
In that example, we have a metric space ($\mathbb{R}^\infty$) which is not $\mathbb{R}^n$ for some $n\in\mathbb N$, and a subset $X$ which is closed, bounded but not compact. There is no way this could contradict the theorem that says that, with respect to the usual distance, every subset of $\mathbb{R}^n$ which is closed and bounded must be compact.