continually increase, but are always less than . Corollary 2: The odd convergents (before the th) continually decrease, but are always greater than . Theorem 5. Corollary 1: A convergent is nearer to the limit of the continued fraction than any fraction whose denominator is less than that of the convergent. Corollary 2: A convergent obtained by terminating the continued fraction just before a large term is a close approximation to the limit of the continued fraction. Semiconvergents If are consecutive convergents, then any fractions of the form where is an integer such that , are called semiconvergents, secondary convergents, or intermediate fractions. The -st semiconvergent equals the mediant of the -th one and the convergent . Sometimes the term is taken to mean that being a semiconvergent excludes the possibility of being a convergent (i.e., ), rather than that a convergent is a kind of semiconvergent. It follows that semiconvergents represent a monotonic sequence of fractions between the convergents (corresponding to ) and (corresponding to ). The consecutive semiconvergents and satisfy the property . If a rational approximation to a real number is such that the value is smaller than that of any approximation with a smaller denominator, then is a semiconvergent of the continued fraction expansion of . The converse is not true, however. Best rational approximations One can choose to define a best rational approximation to a real number as a rational number , , that is closer to than any approximation with a smaller or equal denominator. The simple continued fraction for can be used to generate all of the best rational approximations for by applying these three rules: Truncate the continued fraction, and reduce its last term by a chosen amount (possibly zero). The reduced 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. (See below.) For example, 0.84375 has continued fraction [0;1,5,2,2]. Here are all of its best rational approximations. {| class="wikitable" |- align="center" ! Continued fraction | [0;1] || [0;1,3] || [0;1,4] || [0;1,5] || [0;1,5,2] || [0;1,5,2,1] || [0;1,5,2,2] |- align="center" ! Rational approximation | 1 || || || || || || |- align="center" ! Decimal equivalent | 1 || 0.75 || 0.8 || ~0.83333 || ~0.84615 || ~0.84211 || 0.84375 |- align="center" ! Error | +18.519% || −11.111% ||