Let R denote the set of real numbers. Show that R RR.

(1) Show that the function :R RR given by (x) =(x,x) is injective.

(2) Let f : (0,1)(0,1) (0,1) be defined as follows. Given (x,y) (0,1) (0,1), we write x=0.x₁x₂x_3... and y=0.y₁y₂y₃.... using their infinite decimal expansions (i.e., expansions that do not end with infinitely many zeros) and put

f(x, y) = 0.x₁y₁x₂y₂x₃.....

Show that f is injective.

(3) Use the fact that R (0,1) to obtain an injective function g : RR R.

(4) Use the Cantor-Schroder-Bernstein Theorem to conclude your solution.