The composition of two convex functions is convex
We want to prove that for $x, y \in \Omega$, $(g \circ f)\left(\lambda x + (1 - \lambda) y\right) \le \lambda (g \circ f)(x) + (1 - \lambda)(g \circ f)(y)$.
We have: \begin{align} (g \circ f)\left(\lambda x + (1 - \lambda) y\right) &= g\left(f\left(\lambda x + (1 - \lambda) y\right)\right) \\ &\le g\left(\lambda f(x) + (1 - \lambda) f(y)\right) & \text{(} f \text{ convex and } g \text{ nondecreasing)} \\ &\le \lambda g(f(x)) + (1 - \lambda)g(f(y)) & \text{(} g \text{ convex)} \\ &= \lambda (g \circ f)(x) + (1 - \lambda)(g \circ f)(y) \end{align}