About the set R of reals numbers being not countable.

 Everybody knows that I don't miss an opportunity to make a fool of myself.

That is why I am exposing this problem:

This property (countable) is normally accepted  by proving that the subset of R (real number) , D= [0,1] is not countable.

Nobody would dare to contradict it (Cantor proof!).

But I am on the verge to loose what is left of myself-confidence because I don't find what is wrong in this proof:

First, we know that the set N²= {(n,m), n and m belong to N } is countable . (SCHAUM The algebra of sets Chap 2 p 46).

Next we can map any number that belong to D with N² (it must be the weak part):

0,001035 =1035 10E-6  that can be mapped with (1035, 6) of N².

So how is it that D isn't countable?

What is wrong in my reasoning?

I am not completely dumb so I should soon or later find the answer! 

haqueoui!

We can continue the foolishness with this :

We can construct any element of R by picking an element in N and in D:

if x= 250,001035  the we need  250 in N and 0,001035 in D.

So we can map any element of R with NxD. 

Because D is countable, we can  map it again with (NXN).

And N² being  countable so is R.