I am trying to solve the following exercise. If $x+\frac{1}{x}$ is a natural number, then $x^{n}+\frac{1}{x^{n}}$ is a natural number for all $n\in \mathbb{N}.$
Here what I've done. If $x+\frac{1}{x}$ and $x^{n}+\frac{1}{x^{n}}$ are natural numbers, then $(x+\frac{1}{x})(x^{n}+\frac{1}{x^{n}})\in \mathbb{N}$. But $(x+\frac{1}{x})(x^{n}+\frac{1}{x^{n}})=x^{n+1}+\frac{1}{x^{n-1}}+x^{n-1}+\frac{1}{x^{n+1}}$ and I got that $x^{n+1}+\frac{1}{x^{n+1}}\in \mathbb{N}$. But there is a problem here: I am not supposed to use the second principle of mathematical induction.
My question is: How does one prove that by using the first principle of mathematical induction?
I would appreciate your help.