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A nicer proof of Lagrange's 'best approximations' law?
I was reading through the wikipedia article on continued fractions, and they state, essentially, that for any convergent $\frac{a}{b}$, it is the best approximation you can have. More formally, for an irrational number $x$ with a convergent $\frac{a}{b}$,
$\forall c\forall d \quad |\frac{c}{d}-x| < |\frac{a}{b}-x| \implies d > b$.
However they give no proof of it. Is there a nice one, or did they not give one because it's messy to show?
It's not true. For example, $3/1$ is a convergent of $x = \frac{15}{4} = 3 + 1/(1 + 1/3)$, but it is not a best approximation since $|4/1 - x| < |3/1 - x|$.