I'm trying out a problem I was given and this is the statement:
Prove, or disprove, that every bounded and closed subset of the set of real-valued and bounded functions on [0,1] equipped with the sup norm is compact.
I have a sneaking suspicion that this statement is false but I am unable to find a suitable counterexample. I have proven that the set of real-valued and bounded functions equipped with the sup norm is complete and I have bounded but do not have totally bounded so I believe that I must find a subset that is not totally bounded for my counterexample. My efforts thus far have not been fruitful. Does anyone have any idea how to approach this?