Let f,g be entire functions, with $|f(z)| \leq |g(z)|$ for $|z| > 1,000$ show that $\frac{f}{g}$ is a rational function.

Question

I had my Ph.D qualifying yesterday and they asked me the following question:

Let $f,g$ be entire functions, with $|f(z)| \leq |g(z)|$ for $|z| > 1,000$.

Show that $\frac{f}{g}$ is a rational function.

I know that both $f,g$ has a power series expansion around any point and with infinite radius because each are entire functions. Also we can have that

$|\displaystyle\frac{f}{g}| \leq 1, \forall$ $|z| > 1,000$.

I know I should use the Schwartz lemma in some way but I can not construct the appropriate function such that $h(0) = 0$. Also how can I use the condition of $|z|> 1,000$? Any ideas ? Thanks.