How to prove $f(n) \in O(h(n))$ and $g(n) \in O(i(n))$, then $f(n)+g(n) \in O(max(h(n),i(n)))$


Prove with formal proof that if $ f(n) \in O(h(n))$ and $ g(n) \in O(i(n))$ , then $ f(n)+g(n) \in O(max(h(n),i(n)))$

What I have so far:

By definition of Big-Oh

$ \exists c_1 \exists n_1 \forall n \geq n_1 ( f(n) \leq c_1 \cdot h(n))$

$ \exists c_2 \exists n_2 \forall n \geq n_2 ( f(n) \leq c_2 \cdot i(n))$

Let $ n_0= max(n_1,n_2)$ .

Then $ f(n)+g(n) ≤ c_1 \cdot h(n) + c_2 \cdot i(n) = (c_1 \cdot h + c_2 \cdot i)(n)$ , for all $ n ≥ n_0$ .

I get what the statement is trying to prove, I don’t know how to get $ f(n)+g(n) ≤ c\cdot max(h(n),i(n)) $ , or equivalent, which implies $ f(n)+g(n) \in O(max(h(n),i(n)))$