I will try to match every natural number with every real number between 0 and 1.

Here are all possible combinations of real numbers from 0 to 1,

0.1, 0.11, 0.12, 0.13, 0.14 ...

0.2, 0.21, 0.22, ...

0.3, 0.31, 0.32, ...

0.4, 0.41, 0.42, ...

0.5, 0.51, 0.52, ...

0.6, 0.61, 0.62, ...

0.7, 0.71, 0.72, ...

0.8, 0.81, 0.82, ...

0.9 , 0.91, 0.92, ...

I don't know of a function off the top of my head that will match each natural to each real, but this is how we can imagine them matching. We start with matching 1 to 0.1, then 2 to 0.2, then 3 to 0.3 etc. and go down the column until we get to 9 and 0.9. When we get to 10 we match it with 0.11 and then start all over again. Each natural number can easily match to each real number as the matching goes down the columns.

I would think that this has been thought of before, so where am I going wrong?