Here, the metric space is defined as the set of all bounded sequences, with the distance function defined as the supremum of the absolute value of the difference between corresponding elements.
Intuitively, it seems that this set is not compact, although my intuition for compactness isn't very strong. I've been trying to show that the set doesn't contain all its limit points and is therefore not closed, and hence not compact, but I'm not sure how to proceed. I'm able to show that no element of this set is a limit point, but I'm not sure how to handle limit points outside of the set.
I'd appreciate any advice on how to proceed, or more broadly, intuition for compactness and advice on how to think about compactness in non-traditional metric spaces.