The following is a basic analysis question which I find quite difficult.
Let $\alpha$ be any irrational number. Show that for any positive integer $n$, there exists a ${\bf positive}$ integer $q_n$ and a integer $p_n$ such that
$$ \left| \alpha - \dfrac{p_n }{q_n} \right| < \dfrac{1}{nq_n} $$
And, moreover, show that $\{ p_n \}$ and $\{ q_n \}$ can be chosen in such that we have
$$ \left| \alpha - \dfrac{p_n }{q_n} \right| < \dfrac{1}{q_n^2} $$
Thought:
What I tried to do is to remove the absolute value to have $- 1/nq_n < \alpha - p_n/q_n < 1/nq_n $ and then
$$ \dfrac{n p_n - 1 }{n q_n } < \alpha < \dfrac{ np_n + 1 }{nq_n} $$
This approach leads nowhere.
So, I thought maybe this may help: So, we start with $\alpha $ and we there is a positive integer $q_1$ such that $q_1 \alpha > 1 $ by archimedean property. I am having difficulties here trying to construct $q_n$ and $p_n$. Is this the right strategy?
