Prove that if $f(n) \in \mathcal{O}(h(n))$ and $g(n) \in \mathcal{O}(h(n))$ then $f(n) + g(n) \in \mathcal{O}(h(n))$.
I know that $\mathcal{O}(g(n))=\{f\space | \space\exists c\in\mathbb{R}^{+},\exists n_{0} \in \mathbb{N},\forall n\geq n_{0} : f(n)\leq c\cdot g(n)\}$
However, what do I do with this information to obtain the proof?