I am confused with how you prove this. I understand that both sets can infinitely go on, but how do you prove how many balls will be remaining?

(Mindscape 36, 3.2 of text) Suppose you change the rules of the PingPong Ball experiment discussed in lectures. This time you dump into the barrel 10 Ping Pong balls numbered 110 as before and remove number 1. But next you put in 100 PingPong balls, numbers 11110, and remove number 2. Then you put in 1000 balls, numbered 1111110 and remove number 3, and so on. The question is: How many PingPong balls remain in the barrel after the stopwatch beeps? Innitely many? Finitely many? Can you name one? Explain your answer clearly