An Example with Proof

$6 n \log n + \sqrt{n} \log^2 n = \Theta(n \log n)$.

Need to find n₀, c₁, and c₂, such that

$c_1 n \log n \le 6 n \log n + \sqrt{n} \log^2 n \le c_2 n \log n$

c₁ = 5, c₂ = 7, n₀ = ...

Note: Implications go up. That is, we want to find something at the bottom that will logically imply the statement we're starting with.

So, n₀ = 3 suffices.