Exercise 7.5.3 (Every convergent is a good approximation)

Question

Say that a rational number $a/b$ with $b>0$ is a ``good approximation'' to the irrational number $\xi$ if $$|\xi b -a| = \min_{(x,y) \in \mathbb{Z}^2,\ 0 0$ is a ``good approximation''.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}} ewcommand{\N}{\mathbb{N}} \begin{proof} By the first assertion in Theorem 7.13, if $|\xi y -x | < |\xi k_n - h_n|$, where $n>0,\ (x,y)\in \Z^2$ and $y>0$, then $y > k_n$. Hence, if $n>0$, $$|\xi k_n -h_n| = \min_{(x,y) \in \mathbb{Z}^2,\ 0 0$ is a ``good approximation'' to $\xi$. \end{proof}

Exercise 7.5.3 (Every convergent is a good approximation)

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Exercise 7.5.3 (Every convergent is a good approximation)

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