Big $O$ and Big $\Theta$ proof

Question

Use definitions to prove:

If f and g are nonnegative, and f(n)=$\Theta$(h(n)) and g(n)=$O$(h(n)),

then f(n)+g(n)=$\Theta$(h(n))

I know that

f(n)=$\Theta$(h(n)) means that c|h(n)| <= |f(n) | <= d|h(n)| for some c and d,

and that

g(n)=$O$(h(n)) means that |g(n)|<=d|h(n)| for some d,

But I don't know where to go from there.

Answer 1

For the $\Omega$ bound, note that if

$c|h(n)| \leq f(n)$, then $c|h(n)| \leq f(n) + g(n)$.

For the $O$ bound, note that if

$f(n) \leq c_0|h(n)|$ for all $n \geq n_0$ and $g(n) \leq c_1|h(n)|$ for all $n \geq n_1$, then

$f(n) + g(n) \leq (c_0 + c_1)|h(n)|$ for all $n \geq max(n_0, n_1)$