Range of possible values for nth root of natural number n I'm interested in finding the range of possible values for the $n$th root of a natural number $n \in \mathbb{N}$. Right now, my intuition is telling me that $\forall n \in \mathbb{N}, 1 \leq \sqrt[n]{n} < 2$.
Do you guys think this is correct and if so how would you go about proving it? I guess you could prove that 1 is the infinum and 2 is an upper bound of a set containing all $n$th roots?
Thanks in advance for your help.

As $n\in\Bbb N\implies n\geq1$
Taking $n_{th}$ root on both sides gives, $n^{\frac{1}{n}}\geq 1$
Now, using the fact that $n\lt 2^n\implies n^{\frac{1}{n}}\lt2$
Thus, $1\leq n^{\frac{1}{n}}\lt2$
There might be some tighter bounds.
