Big Theta Proof Tightness

Question

I found that $n_0 = 1 $ and $k=5$ for Big Oh, but I am somewhat confused on how to prove big omega as I have a negative sign in my expression. Furthermore by showing big oh and big omega, am I showing big theta automatically or do I have to take additional steps? Can anyone show me how to formally solve these types of proofs?

Answer 1

Big $\Omega$, $\Theta$, $O$, is, without additional context, an "eventually" criterion.

$$4\cdot n^2 < 5n^2-7n < 5 \cdot n^2$$

is eventually always true as $n\to\infty$.

So $5n^2-7n \in \Theta(n^2)$

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