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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?