Yet another proof of the Weierstrass approximation theorem

Browsing through my Zotero database, I fall upon a paper by Harald Kuhn where he proposes an elementary proof of the Weierstrass approximation theorem. The proof is indeed neat, so here it is. Theorem. — Let $f\colon[0;1]\to\mathbf R$ be a continuous function and let $\varepsilon$ be a strictly positive real number. There exists a polynomial $P\in\mathbf R[T]$ such that…