because I assumed continuous mapping the number c is between a and b it means if it has to be mapped to a natural number the natural number has to be between 22 and 23 but there is no natural number between 22 and 23 , it means c is not mapped to anything
Yeah, OP seems to be assuming a continuous mapping. It still works if you don’t, but the standard way to prove it is the more abstract “diagonal argument”.
Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.
Your explanation is wrong. There is no reason to believe that “c” has no mapping.
because I assumed continuous mapping the number c is between a and b it means if it has to be mapped to a natural number the natural number has to be between 22 and 23 but there is no natural number between 22 and 23 , it means c is not mapped to anything
Yeah, OP seems to be assuming a continuous mapping. It still works if you don’t, but the standard way to prove it is the more abstract “diagonal argument”.
Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.