diff --git a/Advanced/Continued Fractions/parts/02 part B.tex b/Advanced/Continued Fractions/parts/02 part B.tex index 6793528..22fdd67 100644 --- a/Advanced/Continued Fractions/parts/02 part B.tex +++ b/Advanced/Continued Fractions/parts/02 part B.tex @@ -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 $\frac{p_{n+1}}{q_{n+1}}$}