Hi, I am learning about rational and irrational numbers and I came across a question that I am trying to get my head around,
m is a irrational number
n is a rational number which is greater than m
Is it possible to always find a rational number between m and n?
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.
To show this, assume the contrary; there is no such rational number. Let N be a positive integer and consider the numbers where i ranges over the integers. The distance between two consecutive members of this set is . If no member of the set lies between m and n, then we must have, for some i,
Since our choice of N was arbitrary, this inequality must hold for all N. An equivalent inequality is
But we can always find an N such that
by Archimede's principle.
This contradiction shows our assumption that no rational lies between m and n is false.
To follow this we need to agree to just basic facts.
The sum of two rationals is rational. And the positive integers are not bounded above.
Then there is a positive integer such .
From which we get .
Then note that must be rational and .