Let $f\in\mathbb{R}(x)$ be a rational function and $f(x)\in\mathbb{Z}, \forall x\in\mathbb{Z}$, prove that $f$ is a polynomial.

Question

Let $f(x)=\frac{g(x)}{h(x)}$, $g,h\in\mathbb{R}[x]$ be two polynomials and $(g,h)=1$. It's easy to show that $deg(g)> deg(h)$ by taking $x\to \infty$. But since $f(x)\in \mathbb{R}(x)$, for $a\in\mathbb{Z}$, $g(a),h(a)$ may not in $\mathbb{Z}$. Hence it's not easy to say whether $f(a)\in\mathbb{Z}$ when $deg(h)\ge 1$. I don't know how to use the tool of algebra.

Any ideas?