Tonio's proof is that between any two
rational numbers, there exist an irrational number. If x is NOT rational, It's a little more complicated.
Suppose that r and s are the given real numbers, with r< s. Again, look at t= (r+s)/2. That is a
real number between r and s but may not be rational. Let
. There exist an increasing sequence of rational number converging to any real number (The sequence got by truncating the decimal expansion of the number at the
decimal place is such a sequence.) so there exist an increasing sequence of rational numbers,
converging to t. Since it converges to t, there exist N such that if n> N
.
for n> N is then larger than t- (s- r)/2= (s+r)/2- (s- r)/2= r and is less than t< s because it is an increasing sequence.