We showed in class that sqr root 2 is not rational, i.e., that there

is no rational number y such that y^2= 2. The aim of this

problem is to show that there exists a positive real number

y such that y^2= 2.

Let A = {x is in R | x^2< 2}.

(a) Show that A is subset of [−2, 2].

(b) Suppose that y is the least upper bound of A. Show

that

i) y >= 1;

ii)y^2 >= 2;

iii)y^2 <= 2;

iv)y^2 = 2.

[Hint: Use proofs by contradiction for parts ii)–iii).]