Increasing functions form a cone in a vector space

Statement

Suppose $S$ is any subset of the real numbers. Then, the following are true:

1. If $f$ and $g$ are (possibly equal, possibly distinct) increasing functions on $S$, so is $f+g$.
2. If $f$ is an increasing function on $S$ and $a$ is a positive real number, then the scalar multiple $af$ is also an increasing function on $S$.

Put together, these say that increasing functions form a cone in the vector space of functions on $S$.