One of the elementary way to prove of the Brouwer fixed-point theorem is, making it follow from the (smooth) Non-Retraction theorem. The latter is then proven by contradiction by means of a simple computation on the "oriented area" of smooth mappings $g:B\subset \mathbb {R}^n\rightarrow\mathbb {R}^n$ $$\int_B \operatorname{det} D g(x) dx$$

and only involves a differentiation under the sign of integral with respect to the parameter of deformation (I mentioned this proof in this wiki-article) . Due to this fact, I sometimes like to use it in elementary courses as a meaningful application of differential calculus and Lebesgue integration (on the other hand, the geometrical ideas behind remain a bit hidden, but that is an other story).

However, a slight annoyance to me now is, that I can't remember where I read this proof the first time, several years ago. I would be very glad to learn a reference, and (if it is known) the name of the inventor of this nice proof.