I think this problem falls under Real Analysis rather than Pre-Algebra and Algebra, but I think the answer to the question is yes.

My reasoning: from Wikipedia's article on

continued fractions:

"The sequence of rational numbers given by any infinite continued fraction converges to an irrational number, which limit is taken to be the value of the continued fraction. Moreover, every irrational number α is the value of a unique infinite continued fraction, whose coefficients can be found using the non-terminating version of the Euclidean algorithm applied to the incommensurable values α and 1."

Also, further down on the page there is discussion of best rational approximations using either a truncated continued fraction, or else a related continued fraction involving a half-rule (see

here).

Any rate, it follows that rational approximations can get arbitrarily close to irrational numbers, and that there is always a better approximation than any given one, hence it would be between m and n.