
In every closed interval with distinct endpoints, there is at least one rational number and one irrational number.
To prove statement 1, I'll start with some simpler facts:
(2) Between every pair of rational numbers there is an irrational number. And
(3) Between every pair of irrational numbers there is a rational number.
The arguments for these facts depend, in turn, on some more facts:
(4) The sum of rational + rational is rational
proof: a/b + c/d = (ad+bc)/(bd)
(5) The sum of rational + irrational is irrational
proof: if a is rational and b is irrational and a+b=c is rational then c+(a)=b is rational, a contradiction
(6) The product of rational * rational is rational
proof: (a/b)(c/d) = (ab)/(cd)
(7) The product of rational * irrational is irrational as long as the rational is not zero
proof: if a is nonzero rational and b is irrational and ab=c is rational then c(1/a) is rational, a contradiction
Now, let's find an irrational number b that lies between any two rational numbers a and c, where a < c:
Let b = a + (ca)*sqrt(2)/2 is
rational + rational*irrational, so it is irrational.
a < a + (ca)*sqrt(2)/2 < a + (ca) = c, so
a < b < c, and b is irrational, so statement (2) is proven
Now, let's find a rational number b that lies between any two irrational numbers a and c, with a < c:
If ca < 1, then let n = ceiling(log_{10}(ca))
If ca ≥ 1, then let n = 0
Now let a' = a*10^{n+1}, and let c' = c*10^{n+1}
c'  a' ≥ 10, so
a' < ceiling(a') < floor(c') < c', so
a = a'/10^{n+1} < ceiling(a')/10^{n+1} < floor(c')/10^{n+1} <
c'/10^{n+1} = c
Let b = floor(c')/10^{n+1}, so b is a rational number, and
a < b < c, so statement (3) is proven.
Now, let's tackle statement (1): that every closed interval with distinct endpoints contains a rational number and an irrational number.
Let the endpoints be a and c.
If a is rational and c is irrational, or viceversa, then statement (1) is true. Now let's assume they're either both rational or both irrational.
If a and c are both rational, then statement (2) shows there is also an irrational number between them, proving (1).
If a and c are both irrational, then statement (3) shows there is also a rational number between them, proving (1).
In fact, a statement stronger than (1) can be proven:
(8) Every interval, open or closed, that contains more than one point (that is, its endpoints are distinct) contains an infinite number of rationals and an infinite number of irrationals.
Proof: given two real numbers, a and f such that a < f, then let b=(4a+f)/5, c=(3a+2f)/5, d=(2a+3f)/5, and e=(a+4f)/5
Then a < b < c < d < e < f, so the interval between a and f contains two disjoint intervals [b,c] and [d,e]
Since every interval contains two disjoint closed intervals, every interval contains an infinite number of disjoint closed intervals.
Since each closed interval contains a rational number and an irrational number, every interval contains an infinite number of rational numbers and an infinite number of irrational numbers, proving statement (8).
Definition of Interval
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