[An earlier answer by Andrey Rekalo](https://mathoverflow.net/a/60459/8133) suggested the Banach fixed point theorem. I want to elaborate on that answer with a particular direction to take this theorem: *fractals*.

The proof of BFPT is so easy that, without a "flashy" application, it's tempting for students to dismiss it as trivial. But it pairs extremely well with the observation that, if $M$ is a "reasonable" metric space, then appropriate combinations of contraction mappings on $M$ yield interesting contraction mappings on an appropriate hyperspace of $M$. In particular, we have the following:

> $(\star)\quad$ If $F_1,...F_n$ are contraction mappings on the plane $\mathbb{R}^2$ (with the usual metric), then the map $$X\mapsto \bigcup_{1\le i\le n}F_i[X]$$ is again a contraction mapping on the space of compact subsets of $\mathbb{R}^2$ equipped with the Hausdorff metric.

The BFPT then kicks in and says that "solutions exist" to the corresponding equations describing certain sets. It's a fun exercise, then, to describe fractals such as the Sierpinski triangle "algebraically." See e.g. [this survey paper by Natoli](https://math.uchicago.edu/~may/REU2012/REUPapers/Natoli.pdf "Christopher Natoli: Fractals as fixed points of iterated function systems").

In terms of fitting this into 30 minutes, the major hurdle is the definition of the Hausdorff metric, but I think this can be handwaved with a reasonable picture. (And there's no need to prove $(\star)$ in the presentation itself; it can be given as a fun exercise.)