Show that $\max$ function on $\mathbb R^n$ is convex

Fix $k\in \{1,\ldots,n\}$. We have $$\theta x_k + (1-\theta)y_k \leq \theta \max_i x_i + (1-\theta)\max_i y_i$$ because $x_k \leq \max_i x_i$, $y_k \leq \max_i y_i$, $\theta \geq 0$ and $1-\theta \geq 0$.

Since the statement above is true for any $k$ we have:

$$\max_k [\theta x_k + (1-\theta)y_k]\leq \theta \max_i x_i + (1-\theta)\max_i y_i.$$