How can i prove that among any two consecuent convergents to x, al least one of the satisfy $|x-h_{n}/k_{n}|$ $< 1/2k_{n}^2$
I know, by the Thoerem of Hurwitz, that among any three consecutive convegents one of them (al least) satisfy a stronger inequality, but, what if this is the convergent that is out of our selection in the two consecutive for the weak inequality?
Of two consecutive convergents, one is smaller than $x$, the other is greater than $x$, so
$$\left\lvert x - \frac{h_n}{k_n}\right\rvert + \left\lvert x-\frac{h_{n+1}}{k_{n+1}}\right\rvert = \left\lvert \frac{h_n}{k_n} - \frac{h_{n+1}}{k_{n+1}}\right\rvert.$$
Now see that the right hand term is smaller than $\frac{1}{2 k_n^2} + \frac{1}{2k_{n+1}^2}$.