I have four functions $f_1,f_2,g_1,g_2:\mathbb{N}\rightarrow\mathbb{R}^+$. Suppose that $f_1\in\Theta(g_1)$ and $f_2\in\Theta(g_2)$. I need to prove that $f_1+f_2\in\Theta(\text{max}\{g_1,g_2\})$.
This is fairly easy to see with a specific example such as the following:
Let $f_1=x+5$. Then $f_1\in\Theta(x)$.
Let $f_2=2x^2$. Then $f_2\in\Theta(x^2)$.
$f_1+f_2 = (2x^2)+(x+5)$ which is $\in\Theta(x^2)$ and $x^2$ is the max of $x$ and $x^2$.
I cannot figure out how to write this as a general proof.
I can say that $f_1+f_2\in O(\text{max}\{g_1,g_2\})$ since I want a bigger value of $g$.
I am having trouble saying that $f_1+f_2\in\Omega(\text{max}\{g_1,g_2\})$ since I don't necessarily know that with a bigger value of g the inequality of $\Omega$ will still apply.