State and prove Sperner's lemma concerning the colouring of triangles.

Deduce a theorem, to be stated clearly, on retractions to the boundary of a disc.

State Brouwer's fixed point theorem for a disc and sketch a proof of it.

Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous function such that for some $K>0$ we have $\|g(x)-x\| \leqslant K$ for all $x \in \mathbb{R}^{2}$. Show that $g$ is surjective.