Comment by weichi

You need to write down the 1-1 correspondence you are proposing. Don't describe it in words, actually start writing the natural numbers in one column and the corresponding reals in the other. When you do this, you will find either that the 1-1 correspondence that you are envisioning doesn't actually work (i.e., it isn't 1-1), or that it does work ... but in that case, Cantor's diagonalization argument can be applied to it.