The uniform boundedness principle yields a simple non-constructive proof of this fact.
From the en.wikipedia.org
Note that compactness depends only on the topology, while boundedness depends on the metric.
From the en.wikipedia.org
One, boundedness, has already been discussed.
From the en.wikipedia.org
The boundedness reflects the fact that beyond a certain point money ceases being useful at all, as the size of any economy at any point in time is itself bounded.
From the en.wikipedia.org
More examples
Finiteness: the quality of being finite
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. Conversely a set which is not bounded is called unbounded.