It is required to prove that $\lim_{n \to \infty} m^{1/q(n)} = 1$, where $m$ is a natural such that $m \ge 1$ and $\{q(n) \in [1, \infty): n \in \mathbb{N}\}$ diverges to $\infty$.
It has already been proven that $\lim_{n \to \infty} m^{1/n} = 1$ taking any $N > \frac{ln(m)}{ln(\epsilon+1)}$.
But it is not clear how the other hypothesis can be used in a formal way to prove the desired result.
This is what has been tried:
Since $q(n) \in [1, \infty)$, diverges to $\infty$ and $q(n) \neq 0$,
$$\lim_{n \to \infty} \frac{1}{n} = \lim_{n \to \infty} \frac{1}{q(n)}$$
Thus,
$$\lim_{n \to \infty} m^\frac{1}{n} = \lim_{n \to \infty} m^\frac{1}{q(n)}$$
Is the result correct and formal enough?