Interesting but elementary properties of the Mandelbrot Set I suppose everyone is familiar with the Mandelbrot set. I'm teaching a course right now in which I am trying to convey the beauty of some mathematical ideas to first year students. They basically know calculus but not much beyond. The Mandelbrot set is certainly fascinating in that you can zoom in and get an incredible amount of detail, all out of an analysis of the simple recursion $z\mapsto z^2+c$. So my plan is to show them a movie of a deep fractal zoom, and go over the definition of the Mandelbrot set. But I'd like to also show them something mathematically rigorous, and the main interesting properties I know about the Mandelbrot set are well beyond the scope of the course. I could mention connectedness, which is of course a seminal result, but that's probably not that interesting to someone at their level. So my question is whether anyone has any ideas about an interesting property of the Mandelbrot set that I could discuss at the calculus level, hopefully including an actual calculation or simple proof.

A proof that once |z|>2 then the recursion will take it to infinity.
