I have a quick question about best rational approximations of irrational numbers using continued fractions etc. (Continued fraction - Wikipedia, the free encyclopedia) Suppose we have a real number x, and we know that some rational number \frac{a}{b} is a best approximation of x, in other word a convergent of the continued fraction of x.

Now suppose some other ration number \frac{a'}{b'} is such that x<\frac{a'}{b'}<\frac{a}{b}. We then know for sure that b<b', but is \frac{a'}{b'} necessarily a best approximation of x as well? It seems to me like this would be true, but I can't think of any good reason why.