# Exercise 3.2.1

Show that if $f(n)$ and $g(n)$ are monotonically increasing functions, then so are the functions $f(n) + g(n)$ and $f(g(n))$, and if $f(n)$ and $g(n)$ are in addition nonnegative, then $f(n) \cdot g(n)$ is monotonically increasing.

So:

$$f(m) \leq f(n) \quad \text{for } m \leq n \\ g(m) \leq f(n) \quad \text{for } m \leq n \\$$

When we combine them, we get:

$$f(m) + g(m) \leq f(n) + g(n)$$

Which proves the first function.

Then:

$$f(g(m)) \leq f(g(n)) \quad \text{for } m \leq n$$

This is true, since $g(m) > g(n)$ and $f(n)$ is monotonically increasing.

If both functions are nonnegative, then we can multiply the two inequalities and we get:

$$f(m) \cdot g(m) \leq f(n) \cdot g(n)$$