# Prove Θ(n) + O(n^2) ≠ Θ(n^2)

How would I go about proving this statement?

Θ(n) + O(n^2) ≠ Θ(n^2)


I know how to prove if given a function f(n) if it's big o but I do not understand how to go about this type of problem.

• Welcome to Computer Science! What have you tried? Where did you get stuck? We do not want to just hand you the solution; we want you to gain understanding. However, as it is we do not know what your underlying problem is, so we can not begin to help. See here for tips on asking questions about exercise problems. If you are uncertain how to improve your question, why not ask around in Computer Science Chat? – Raphael Sep 6 '16 at 8:21
• What if you start by writing down the definition of Θ(n) and O(n)? Once you do this, the answer is blatantly obvious. – gnasher729 Sep 6 '16 at 8:26

$$f(n) = n + n$$
Clearly $f(n) \in Θ(n) + O(n^2)$. However, $\frac{f(n)}{n^2} = 2/n$, which shows that $f(n)\not \in Θ(n^2)$.