I am reading the proof for the following theorem:

The setsandare 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.