My favorite example of real analysis' application in micro is the proof that lexicographic preference cannot be represented by any utility function. (A succinct version of the proof is given in p.43 of MWG.) It uses the notion of cardinality of sets, in particular, the different cardinalities between the rationals and the reals, which are some of the first things one learns in a real analysis course.
Other materials in real analysis, such as set operations, closedness, openness, continuity, differentiability, boundedness, compactness, etc., find their ways into every nook and cranny of economic theory. Some examples include:
- Classical demand theory, which deals mostly with $\mathbb R^n$, the $n$-dimensional commodity space, draws on a lot results from real analysis
- Any theoretical work that involves probability, e.g. econometrics as well as risk/uncertainty in micro or macro, would rely on some basic measure theory, an advanced topic in real analysis
- Dynamic programming/optimal control, a technique for solving DSGE (and other) models, relies on the contraction mapping theorem
- Separating hyperplane theorem is used to prove the existence of general equilibrium
- Brouwer's fixed point theorem is used to prove the existence of Nash equilibrium
See some of the book recommendations and related questions on this site (1, 2, 3, 4) for more examples.