(Hard) Construct an explicit bijection from (0,1] to (0,1). Hint: One approach is as follows: First map (1/2,1] to (0,1/2], then map (1/4,1/2] to (1/2,3/4], etc. Write down the map explicitly, that is, write down an algorithm that tells you exactly what number goes where. Then prove that the map is a bijection.