The positive rational numbers are countable ie they can be listed. Take a particular list of the positive rationals s
1, s
2, s
3...
Now sort this list into ascending order. Let's look at the 10,000 th and the 10,001 th elements in the sorted list. Call them r
10,000 and r
10,001.
Since r
10,000 and r
10,001 are not the same they must differ by a non-zero number. If we write r
10,000 as p
1/q
1 and r
10,001 as p
2/q
2 then their difference is p
2/q
2 - p
1/q
1 = (p
2q
1 - p
1q
2)/q
1q
2. Now the smallest number that the numerator can be is 1 (note that p
2/q
2 > p
1q
1). Consider the number 1/2q
1q
2. It must be smaller than the difference between r
10,000 and r
10,001. If we add 1/2q
1q
2 to r
10,000 then we get another rational number (the sum of two rationals is always another rational). Call this r.
So r is a rational number and lies between our two numbers. Yet r
10,000 and r
10,001 are consecutive rationals, ie no rational number can lie between them.
Solution