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,

$m \le n \implies f(m) \le f(n) \tag {1}$ $m \le n \implies g(m) \le g(n) \tag {2}$

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 we can multiply inequalities (1) and (2), to say:

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

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