PROOF Please state what type of proof youre using to solve a

PROOF. Please state what type of proof you\'re using to solve as well as a clear detailed solution. Thank you.

Find a bijection from [0,1) into (0,1). Prove that the function you found is bijective. PROOF. Please state what type of proof you\'re using to solve as well as a clear detailed solution. Thank you.

Solution

Choose an infinite sequence (x_n)n?1 of distinct elements of (0,1). Let X={x_n?n?1}, hence X?(0,1). Let x0=1. Define f(x_n)=x_n+1 for every n?0 and f(x)=x for every x in (0,1)?X. Then f is defined on (0,1] and the map f:(0,1]?(0,1) is bijective.

To sum up, one extracts a copy of N from (0,1) and one uses the fact that the map n?n+1 is a bijection between N?{0} and N.