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.

\(f(n)\) and \(g(n)\) being monotonically inceasing functions tells us that \(m \leq n \implies f(m) \leq f(n)\) and \(g(m) \leq g(n)\). We are able to combine these two inequalities as \(f(m) + g(m) \leq f(n) + g(n)\) which tells us that \(f(n) + g(n)\) is also monotonically increasing.

Because \(m \leq n \implies g(m) \leq g(n)\) then we also know that \(f(g(m)) \leq f(g(n))\) which tells us that \(f(g(n))\) is monotonically increasing as well.

If \(f(n) \geq 0\) and \(g(n) \geq 0 \ \forall \ n\) then we are also able to multiply the inequalities as \(f(m) \cdot g(m) \leq f(n) \cdot g(n)\) which demonstrates that \(f(n) \cdot g(n)\) is monotonically increasing if both functions are nonnegative.