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Mark 2023-10-26 09:43:16 -07:00
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Advanced/Continued Fractions/parts/02 part B.tex
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@ -247,7 +247,7 @@ $\frac{p}{q}$ satisfying $|\alpha - \frac pq| < \frac{1}{2q^2}$.
\item Prove that $(x+y)^2 \geq 4xy$ for any real $x,y$.
\item Let $p_n/q_n$ be the $n$th convergent to $\alpha$. Prove that
\[
|\frac{p_n}{q_n} - \frac{p_{n+1}}{q_{n+1}}|^2 \ \geq \ 4 | \frac{p_n}{q_n} - \alpha | | \frac{p_{n+1}}{q_{n+1}} - \alpha |
\biggl|\frac{p_n}{q_n} - \frac{p_{n+1}}{q_{n+1}}\biggr|^2 \ \geq \ 4 \biggl| \frac{p_n}{q_n} - \alpha \biggr| \biggl| \frac{p_{n+1}}{q_{n+1}} - \alpha \biggr|
\]
\hint{$\alpha$ lies in between $\frac{p_n}{q_n}$ and $\frac{p_{n+1}}{q_{n+1}}$}