There must be a flaw in my thinking below. Can someone please help by pointing it out? Thanks.

Theorem: The reals in the interval are a countable set.

Proof: Show that a one-to-one mapping exists between the reals in the interval and the natural numbers. Any real number
r={x: xeR, xe} has a decimal representation given by:
r= Sum(Cn 10^-n), where n = 1 to infinity, and Cn = {0,1,...9}.

Construct the corresponding natural number:
m=Sum(Cn 10^(n-1)), where n = 1 to infinity,
and Cn = {0,1,...9}.

For any real number, r, there exists a corresponding natural number, m. This correspondence provides a one-to-one mapping of the reals in the interval onto the natural numbers. Therefore, by definition, the real numbers in the interval form a countable set.

Clearly, this is not true -- but what's wrong?!! A different, but related question is why can't we construct a counter- arguement for the uncountablity of the natural numbers in a way similar to the Cantor arguement for the uncountability of the reals (i.e., no list can contain a complete denumeration of the reals -- a procedure is given for constructing one not already on the list). Isn't the same thing true of the natural numbers? -- crudely thinking of them as a reflection of the reals in "onto the other side" of the decimal point?