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I'm trying to solve the following problem but I'm having some difficulties..

Let $a_0,a_1,\dots,a_n$ and $b_0,b_1,\dots,b_n,b_{n+1}$ be positive integers. Give conditions that make the following statement true: $$[a_0;a_1,\dots,a_n]<[b_0;b_1,\dots,b_n,b_{n+1}]$$

I thing a have to use convergents of the continued fraction but I don't know how. Thanks!

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Hints:

  1. Left side is between $a_0$ and $a_0 + 1$.
  2. $[a_0; a_1, \ldots ] = a_0 + 1/[a1; a_2, \ldots]$
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  • $\begingroup$ can you be a little more specific please. $\endgroup$ – ViKaN-HND504 Nov 6 '14 at 12:05
  • $\begingroup$ If $a_0 \ne b_0$, that decides it. Otherwise, look at $[a_1;a_2,\ldots]$ and $[b_1;b_2,\ldots]$. $\endgroup$ – Robert Israel Nov 6 '14 at 16:01

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