Question

Use the results of exercises 18 and 19 to prove that given any two rational numbers r and s with r , s, there is another rational number between r and s. An important consequence is that there are infinitely many rational numbers in between any two distinct rational numbers.

Answer 1

Answer:

Step-by-step explanation:

Let r and s be two rational numbers

Without loss of generality assume that r<s, because one number has to be necessarily less than the other otherwise two would be equal.

Then find mid value of r and s as

$\frac{r+s}{2} =t$

So we have atleast one rational number between r and s. Note that t is rational because it is sum of two rational numbers r/2 and s/2

Now using r and t we find one rational number say u between r and t.  

Again with r and u we find another rational number between them

This process can be repeated infinitely

Thus we conclude there are infinitely many rational numbers in between any two distinct rational numbers.