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.

Fix $n\geq 1$.
By basic axioms of $\mathbb{R}$, you can prove there is a unique $y>0$ such that $y^n=n$.
Note that the function $y\longmapsto y^n$ is increasing.
Prove by induction that $2^m>m$ for all $m\geq 1$.
In particular, $2^n>n$.
Conclude that $y<2$.
