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.

As $f(n)$ and $g(n)$ are monotonically increasing functions,

$m \le n \Rightarrow f(m) \le f(n) \tag 1$ $m \le n \Rightarrow g(m) \le g(n) \tag 2$

Therefore, $f(m) + g(m) \le f(n) + g(n)$, i.e. $f(n) + g(n)$ is monotonically increasing.

Also, combining (1) and (2), $f(g(m)) \le f(g(n))$ Therefore, $f((g(n))$ is also monotonically increasing.

If $f(n)$ and $g(n$) are nonnegative then combining this new information with (a) and (2), we can say:

Therefore, $f(n) \cdot g(n)$ is monotonically increasing.