TITLE: Applications of model theory to analysis
QUESTION [29 upvotes]: Some of the more organic theories considered in model theory (other than set theory, which, from what I've seen, seems to be quite distinct from "mainstream" model theory) are those which arise from algebraic structures (theories of abstract groups, rings, fields) and real and complex analysis (theories of expansions of real and complex fields, and sometimes both).
While relationships with algebra seem quite apparent, I wonder what are some interesting results in real and complex analysis that have nice model-theoretical proofs (or better yet, only model-theoretical proofs are known!)? 
Of course, there's nonstandard analysis, but I hope to see some different examples. That said, I wouldn't mind seeing a particularly interesting application of nonstandard analysis. :)
I hope the question is at least a little interesting. I have only the very basic knowledge of model theory of that type (and the same applies to nonstandard analysis), so it may seem a little naive, but I got curious, hence the question.

REPLY [4 votes]: Using continuous model theory, Ilijas Farah, Brad Hart and David Sherman proved `blind man's version' of the Connes Embedding Problem:

There exists a separable ${\rm II}_1$-factor $M$ such that every other separable ${\rm II}_1$-factor embeds into an ultrapower of $M$.

I find this result pretty amusing. On the C*-algebra side, I like results à la the Löwenheim–Skolem theorem: build a non-separable C*-algebra with your favourite axiomatisable properties and go to a separable C*-algebra which inherits those properties.