Let $\alpha$ be any real number such that $ \alpha + \frac{1}{\alpha} \in \mathbb{Z}$. Prove that $\alpha^{n} + \frac{1}{\alpha^{n}} \in \mathbb{Z}$ for any $n \in \mathbb{Z}$

My attempt was to try induction but first I realized that $\alpha$ has to be $ \pm1$ in order for it be be an integer. 

Base case $n=1$ then $ \pm1^{1}+ \frac{1}{\pm1^{1}} \in \mathbb{Z}$

Assume $P(k): \alpha^{k} + \frac{1}{\alpha^{k}} \in \mathbb{Z}$ is true for some $k \geq 1$ I want to show that $P(k+1)$ is true. 

$\alpha^{k+1} + \frac{1}{\alpha^{k+1}}=\alpha^{k}\*\alpha+\frac{1}{\alpha^{k}\*\alpha}$ so I get an integer multiplied by $\pm1$ so it is an integer. Thus the statement is true for all $n$

Any clarification would be helpful.