Have you taken a look at some texts on functional analysis? These texts will have a lot of proofs for things in infinite-dimensional space which should prove quite challenging.
I'm looking for problems catered to the undergrad level in real analysis concerning difficult limits of functions and difficult proofs pertaining to continuity of functions. All the time I've thus spent scouring the internet for "difficult limit problems" has mostly just turned up epsilon delta proofs exposed supplementary to an elementary calculus class or nothing beyond what is already in my text (Bartle and Sherbert; basically at chapter five for those of you familiar with it). Is there anywhere to find problems that require perhaps a "clever insight" to everything working out nicely, or involve slightly more ugly or slightly beyond the ordinary functions, or things like that?