State and prove Sperner's lemma concerning colourings of points in a triangular grid.

Suppose that $\triangle$ is a non-degenerate closed triangle with closed edges $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$. Show that we cannot find closed sets $A_{j}$ with $A_{j} \supseteq \alpha_{j}$, for $j=1,2,3$, such that

$$\bigcup_{j=1}^{3} A_{j}=\Delta, \text { but } \bigcap_{j=1}^{3} A_{j}=\emptyset$$