If $f\in H(\mathbb{C})$ and $f(n^{\frac{1}{n}})\in\mathbb{R}$ for every $n\in\mathbb{N}$, show that $f(\mathbb{R})\subset\mathbb{R}$.

Question

I want to show that if $f\in H(\mathbb{C})$ and $f(n^{\frac{1}{n}})\in\mathbb{R}$ for every $n\in\mathbb{N}$ then $f(\mathbb{R})\subset\mathbb{R}$. I think that I should use the identity principle ( or identity theorem ) but I cant figure out how exactly to do it. The set A=$\{n^{\frac{1}{n}}:n\in\mathbb{N}\}$ has an accumulation point which is 1 and is contained in $\mathbb{R}$ and since f is analytic then it is holomorphic on the real axis, so $f(x)=g(x)$ for every $x\in\mathbb{R}$ by the identity principle but I think that this explanation is not that good. Any ideas of how to give better proof?