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Let $\alpha = [a_0;a_1,a_2,\ldots] \in \mathbb{R}$ and $(p_n/q_n)$ be the convergents to the continued fraction of $\alpha$. Prove that, if $q_n \leq q < q_{n+1}$, gdc($p$,$q$)$=1$ and $p/q \not = p_n/q_n$ then $|\alpha - p/q| \leq |\alpha - p_n/q_n|$ only if $\frac{p}{q} = \frac{p_{n+1}-rp_n}{q_{n+1}-rq_n}$, where $r \in \mathbb{N}$.

This is a problem I can't solve. I was only able to show that the continiued fraction of $p/q$ begins with $a_0, a_1, \ldots, a_n$. My guess is that the next step is showing that it actually ends at the $(n+1)$th term, and that, I think, would solve it.

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  • $\begingroup$ As a general comment, Alan Baker's "Concise introduction to the Theory of Numbers" is excellent on this and much more besides. $\endgroup$ – Richard Martin Nov 26 '18 at 17:56

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