# what would be a example in a general metric space about closedness and boundedness not imply sequential compactness?

Unlike in $R^n$, closed and bounded doesn't guarantee sequential compactness. Textbook examples includes sup metric and R^infinite metric. I am wondering what would be a example of closed and bounded doesn't imply sequential compactness in more general Metric space $(X,\rho)$?

Take the space $l_2$ (ie, $d(x,y) = \sqrt{\sum_n |[x]_n - [y]_n|^2}$) and the set $C = \{ e_n \}_n$, where $e_n$ is the $n$th unit vector (ie, zeros everywhere except for the $n$th position which is one).
Then $C$ is bounded, closed but not compact (or sequentially compact).
Take any metric space $(X,\rho)$ which is not sequentially compact (= compact, for metric spaces), and replace $\rho$ with the standard bounded metric $\rho_1(x,y) = \min(\rho(x,y),1))$. Then $X$ itself is closed and $\rho_1$-bounded but not compact, despite the fact that the induced topology $\tau_{\rho_1}$ is the same as the induced topology $\tau_{\rho}$ of $(X,\rho)$.