Complexity of $n \sqrt{n}$

Question

I know that $O(n\sqrt{n}) = O(n)$ but I have no idea how I can prove this. Can you give me a hint?

Answer 1

What you want to prove is actually not correct.

Obviously, $n\sqrt{n} \in O(n\sqrt{n})$, but

$$\frac{n\sqrt{n}}{n} = \sqrt{n}.$$

Since $\sqrt{n}$ is not bounded, we have $n\sqrt{n} \notin O(n)$.