Proposition 13.1.4.label Let $E$ be a vector space over $\real$ and $A \subset E$ be convex, then:

1. (1)

   For any $f, g: A \to (-\infty, \infty]$ convex and $\lambda \ge 0$, $\lambda f + g$ is convex.

2. (2)

   For any convex functions $\cf \subset (-\infty, \infty]^{A}$, $\sup_{f \in \cf}f$ is convex.