Let $I$ be an interval of $\mathbf{R}$. Let $f : I \rightarrow \mathbf{R}$ be a convex function. Show that, for all $p \in \mathbf{N}^\star$, for all $(\lambda_1, \ldots, \lambda_p) \in (\mathbf{R}_+)^p$ such that $\sum_{i=1}^p \lambda_i = 1$ and for all $(x_1, \ldots, x_p) \in I^p$, we have:
$$f\left(\sum_{i=1}^p \lambda_i x_i\right) \leq \sum_{i=1}^p \lambda_i f(x_i)$$
Hint: You may proceed by induction on $p$.