Question

1. .Show that the following statements are true. You may assume the properties of real numbers in Appendix A.
   1. If r and s are rational numbers, then (r+s)/2 is rational.
   2. For all real numbers a and b, if a < b then a < (a+b)/2 < b
2. Based on a, b above, 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 of this is that there are infinitely many rational numbers between any two rational numbers.

Answer 1

a. Rational numbers are quotients of integers, which can be ratio of two integers or single fraction, therefore here, we can say that  r and s are rational numbers.

r = c/d and s = e/f for some intergers, c,d,e and f where d != 0 and f!=0

so proof that r and s are rational numbers.

here given that (r+s ) / 2 is rational.

also, r + s is rational

ie. r =1 and s = 2

so, 1+ 2 /2 = 3 / 2

therefore, (r + s) /2 is rational.

b. If a < b then a < (a+b/)2 <b

here, a < b and we need to get a+b/2

so add b for both sides

a + b < b +b ie. a+b < 2b

so, we have a+b / 2 < b

therefore, here we get a + b/2 < b

other side, a < b

add a both sides a+a = 2a

ie. 2a < a+b

we get, a < a+b/2

second side,
a < b

2a < a + b

in second case, we get a < a+b/2

So, from both cases we get a < (a+b)/2 <b

ie. means a is less than (a+b)/2 and also (a +b)/2 is less than b

Similarly, two rational numbers r and s with r < s

ie. means we know given any two distinct numbers because of the distinct one of the bigger than other, we can find numbers between them