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 , and we know that some rational number is a best approximation of , in other word a convergent of the continued fraction of .
Now suppose some other ration number is such that . We then know for sure that , but is necessarily a best approximation of as well? It seems to me like this would be true, but I can't think of any good reason why.