Let $n \in \mathbb{N}^{*}$. We denote by $\chi_{n} : \mathbb{R} \rightarrow \mathbb{R}$ the continuous function that equals 1 on $[-n, n]$, equals 0 on $]-\infty, -n-1] \cup [n+1, +\infty[$ and is affine on each of the two intervals $[-n-1, -n]$ and $[n, n+1]$.
Show that:
$$\forall x, y \in \mathbb{R},\quad \chi_{n}(x)^{\lambda} \chi_{n}(y)^{1-\lambda} \leq \chi_{n+1}(\lambda x + (1-\lambda) y)$$