I am reading the proof for the following theorem:
The sets and are numerically equivalent.
I understood the part where the author proved the existence of the one-to-one function , but I find it hard to understand the converse; i.e., the proof of the existence of the one-to-one function .
It says as follows :
and it continues to show that is one-to-one.We define a function . For , define , where
Thus is a real number in (0,1), whose decimal expansion consists only of 1s and 2s.
Question: Let's say . What do the first two numbers look like? Please show examples.
is just some point that is in or (I assumed WLOG above that ) which isn't in the other. We know there must be at least one since the two sets aren't equal.