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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!

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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.

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    From my understanding, if both $f(n)$ and $g(n)$ are asymptotically slower than $c_1n$ and $c_2n$, then that means adding the functions together would result in $f(n) + g(n)$ being asymptotically slower than $c_1n + c_2n$. Am I'm thinking correctly? – 2017-01-11
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    @Jasmine Yes. The formal way to state this is by saying that $a – 2017-01-11