Solution
= Solution
For $0\leq t\leq1$ and $x,y\in\mathbb R^d$, <convex function>[convexity] gives
$$
f_i(tx+(1-t)y)\leq tf_i(x)+(1-t)f_i(y).
$$
Taking the maximum over $i$ and then bounding each term by the corresponding endpoint maxima yields
$$
\max_i f_i(tx+(1-t)y)
\leq t\max_i f_i(x)+(1-t)\max_i f_i(y).
$$
\b[Thus the finite <pointwise maximum of convex functions> is convex. Summing the original inequalities over $i$ proves that $\sum_i f_i$ is convex as well.]