If a sequence of rational numbers converges to irrational number do the numerator and denominator

Question

I’m attempting to solve the following question: $a_n$ is a sequence of rational numbers, for better understanding let’s write it as: $p_n/q_n$ (where $p<n$ and $q_n$ are sequences over the naturals $\Bbb N$). If the limit of an (when $n$ diverges (to infinity)) is $r$ which is an irrational number : $$\lim a_n = r| r\in \Bbb R\setminus\Bbb Q$$ Does that mean that $p_n$ and $q_n$ both diverge (to infinity)?

I’ve found many examples that support this theorem but examples aren’t proof so I’m not sure if it’s actually true and why? Is there a way to prove it or is it just a known fact?

Answer 1

If both $p_n$ and $q_n$ are bounded, then, by virtue of them being integers, there are only a finite number of values that $(p_n, q_n)$ can assume. So $\frac{p_n}{q_n}$ can only assume a finite number of values (all rational numbers), which means that the limit $r$ has to be rational (and in fact, the sequence is constant and equal to $r$ for large values of $n$).

If exactly one of $p_n$ or $q_n$ is bounded, then $r$ is either $0$ or $+\infty$, neither of which is an irrational.

Conclusion: Both have to be unbounded.

As @orangeskid said, since every subsequence is unbounded, the sequences tend to infinity.

Answer 2

Both $\ P=\{p_n:n\in\mathbb{N}\}\ $ and $\ Q=\{q_n:n\in\mathbb{N}\}\ $ are bounded below by $\ 1.$

If $\ P\ $ is bounded above and $\ Q\ $ is not bounded above, then $\ \displaystyle\lim_{n\to\infty}\frac{p_n}{q_n}=0,\ $ which is not irrational.

If $\ P\ $ is bounded above and $\ Q\ $ is bounded above, then the number of different fractions that $\ \frac{p_n}{q_n}\ $ can be is $\ \leq\ \max(P) \max(Q),\ $ which is finite (under the assumption that $\ P\ $ and $\ Q\ $ are both bounded above). The limit of a sequence of a finite amount of different rational numbers cannot be irrational.

If $\ P\ $ is not bounded above and $\ Q\ $ is bounded above then $\ \displaystyle\lim_{n\to\infty}\frac{p_n}{q_n}\ $ is not a finite number and therefore cannot be irrational.

If $\ P\ $ is not bounded above and $\ Q\ $ is bounded above then it is possible for $\ \displaystyle\lim_{n\to\infty}\frac{p_n}{q_n}\ $ to be irrational.