I have read this from here
The simple continued fraction for $x$ generates all of the best rational approximations for $x$ according to three rules:
Truncate the continued fraction, and possibly decrement its last term.
The decremented term cannot have less than half its original value.
If the final term is even, half its value is admissible only if the corresponding semiconvergent is better than the previous convergent.
It means the best rational approximations of $x$ are the convergents and some semi-convergents of the continued fraction. But I can't find the proof anywhere. How can I prove it?
The question was already asked here, but it doesn't have any answer.