# Question: 2 is not a rational number consider the following...

###### Question details

√ 2 is not a rational number. Consider the following sets A = {x ∈ Q | x ≤ √ 2} and B = {x ∈ Q | x ≥ √ 2}. Our goal in this exercise, is to show that set A has no maximum element and no least upper bound within the set of rational numbers. The following steps will be useful in proving these facts.

(a) Assume that q ∈ A is the maximum element of A. Show that it must be the case that q' = 2/q is the smallest element of set B.

(b) Show that it must be the case that q' > q.

(c) Consider the number r = (q+q')/2 . Show that either r ∈ A or r ∈ B.

(d) Show that either option in the previous step contradicts the assumption that q is the maximum element of A.

(e) Show that the set B has no least element.

(f) Show that within the set Q, the set A has no least upper bound.