calculus - How to prove $lim_{n to infty} sqrt{n}(sqrt[n]{n} - 1) = 0$?

I want to show that $$\lim_{n \to \infty} \sqrt{n}(\sqrt[n]{n}-1) = 0$$ and my assistant teacher gave me the hint to find a proper estimate for $\sqrt[n]{n}-1$ in order to do this. I know how one shows that $\lim_{n \to \infty} \sqrt[n]{n} = 1$, to do this we can write $\sqrt[n]{n} = 1+x_n$, raise both sides to the n-th power and then use the binomial theorem (or to be more specific: the term to the second power). However, I don't see how this or any other trivial term (i.e. the first or the n-th) could be used here.

What estimate am I supposed to find or is there even a simpler way to show this limit?

Thanks for any answers in advance.

Answer

The OP's attempt can be pushed to get a complete proof.
$$
n = (1+x_n)^n \geq 1 + nx_n + \frac{n(n-1)}{2} x_n^2 + \frac{n(n-1)(n-2)}{6} x_n^3 > \frac{n(n-1)(n-2) x_n^3}{6} > \frac{n^3 x_n^3}{8},
$$

provided $n$ is "large enough" ¹. Therefore, (again, for large enough $n$,) $x_n < 2 n^{-2/3}$,
and hence $\sqrt{n} x_n < 2n^{-1/6}$. Thus $\sqrt{n} x_n$ approaches $0$ by the sandwich (squeeze) theorem.

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¹In fact, you should be able to show that for all $n \geq 12$, we have
$$
\frac{n(n-1)(n-2)}{6} > \frac{n^3}{8} \iff \left( 1-\frac1n \right) \left( 1- \frac2n \right) \geq \frac34.
$$