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 r10,000 and r10,001.
Since r10,000 and r10,001 are not the same they must differ by a non-zero number. If we write r10,000 as p1/q1 and r10,001 as p2/q2 then their difference is p2/q2 - p1/q1 = (p2q1 - p1q2)/q1q2. Now the smallest number that the numerator can be is 1 (note that p2/q2 > p1q1). Consider the number 1/2q1q2. It must be smaller than the difference between r10,000 and r10,001. If we add 1/2q1q2 to r10,000 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 r10,000 and r10,001 are consecutive rationals, ie no rational number can lie between them.