Prove asymptotic relationship using the limit method

Question

Prove that $$n\log(n) = o(n^{3/2})$$ using the limit method`

Note that log is in base 2.

I've missed a few classes due to illness and am trying to catch up. From the notes, I see that I can compute the following:

$$\lim_{n\to \infty} \dfrac{n\log(n)}{n^{3/2}} = 0$$

But now what? Has it been proved? It makes sense that $n^{3/2}$ bounds $n\log(n)$ however, let's also note that this is small-oh not big-oh. Am I finished the proof?

Thanks a lot in advance.

Answer 1

Yes you have finished the proof.

$g(n) = o(f(n))$ is defined to mean $\lim_{n\to +\infty}\dfrac{g(n)}{f(n)} = 0$