real analysis - Verification of $lim_{n rightarrow infty} sqrt[n]{n^3}=1$

I am interested in the limit

$$\lim_{n \rightarrow \infty} \sqrt[n]{n^3}$$

Can we simply conclude that:

$$\lim_{n \rightarrow \infty} (\sqrt[n]{n})^3= 1^3=1.$$

I have proven that $\sqrt[n]{n}\rightarrow1$ earlier in this textbook. Also since the limit of a power is the power of the limit.

Answer

Yes, we can simply do that. Since the exponent $^3$ is a constant neutral number (meaning we may interpret it as a fixed number of multiplications) we can move the limit inside of it. So if you already know $\lim_{n\to\infty}\sqrt[n]n=1$ then that's a full proof.