Infinite dimensional constructions, such as spaces of diffeomorphisms, spectra, spaces of paths, and spaces of connections, appear all over topology. I rather like them, because they sometimes help me to develop a good mental picture of what is going on. Emmanuel Farjoun, in the first lecture of an Algebraic Topology course about a decade ago, described "becoming comfortable with the idea of infinite dimensional manifolds" as being "one of the main conceptual advances in topology in the latter half of the 20th century". But, as I realized in a discussion yesterday, I don't understand whether infinite dimensional spaces are needed, or whether they are merely an intuitive crutch.
Are there situations in which a significant finite dimensional result strictly requires an infinite dimensional construction in order to prove it or in order to properly understand it?
If the answer is no, then at least are there important finite-dimensional theorems for which infinite-dimensional proofs are "clearly" the easiest and the most natural? (for some conceptual reason which you can explain; not just "a finite-dimensional proof isn't known yet").
A closely related question is this, although its focus is somewhat different, and none of the answers there apply here; however Andrew Stacey's answer which argues that infinite-dimensional constructions are usually not strictly necessary, is relevant.
Edit: Your favourite finite-dimensional result proved by infinite-dimensional means answers this question only if you can explain to me why I should not expect a finite-dimensional proof to exist or to be anywhere near as "good".