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"initial segments of the continued fraction lead to the best rational approximations"

Yow, are you a lawyer yet?  If "best rational approximations" are those which
are closer than any with smaller denominator, then your "leads to" is misleading,
since reducing the final term of a truncated cf by less than half its correct
value still gives a "best" approximation.  Reducing by fully one half is
"best" in only about half of the (even) cases.  The precise terms and conditions
are intelligible by anyone who has recently passed a Bar exam, and I will inflict
them only on the avowedly curious.