If $f(n) \in \Theta(n)$ and $g(n) \in \Theta(n)$, then $f(n) + g(n) \in \Theta(n)$
I'm supposed to prove the following theorem below using the definition of $\Theta(n)$. I know the definition of Big-Theta, but I don't understand how to use it prove the theorem. I'd prefer to NOT be given a full proof to this theorem as I'd like to learn how to do this myself. But if possible, any guidance on how to start this would be great!
Start with the definition: There are constants $a_1$ and $a_2,$ $b_1$ and $b_2,$ such that $$ a_1n < f(n) < a_2n $$ $$ b_1n < g(n) < b_2n $$ for sufficiently large $n$.
Now you want to say something about $f(n) +g(n).$ Can you use the above inequalities to to get an inequality for $f(n)+g(n)$? See if you can put it into the form where it fits the definition of $\Theta(n)$ for some new constants $c_1$ and $c_2$ that you can derived from previously defined quantities.