Let $(a_n)_{n\in\mathbb{N}}$, $(b_n)_{n\in\mathbb{N}}$ two sequences of integers.
Prove that if $u_n=\frac{a_n}{b_n}$ converges to an irrational $x$, then both $a_n$ and $b_n$ diverge to infinity.
I started with a proof by contradiction and proved the first case:
- If $a_n$ and $b_n$ converge to $l$ and $l'$ respectively, then $x=\frac{l}{l'}$ which is a contraction.
Now, I have to prove it when one or both diverges.