Asymptotic Notation – does n (log n) (log n) simplify?

Here’s a proof by contradiction:

Let’s say that a function f(n) = n(log n)(log n). Assume that we think it’s also Θ(n log n), theta(n log n), so in other words there is an a for which f(n) <= a * n log n holds for large n.

Now consider f(2^(a+1)):

f(2^(a+1)) = 2^(a+1) * log(2^(a+1)) * log(2^(a+1)) = 2^(a+1) * log(2^(a+1)) * (a+1), which is clearly larger than a * 2^(a+1) * log(2^(a+1)), and we have a contradiction. therefore f(n) is not an n log n function.