Odd numbers

Let's imagine we have two bags. In one bag, let's call it N-bag we put numbered balls, in the sequence 1,2,3,...; in the other bag, let's call it E-bag, we put only the even numbered balls 2,4,6,...; the procedure is as follow:

step 1: we put ball-1 in N-bag
step 2: we put ball-2 in N-bag and another ball-2 in E-bag
step 3: we put ball-3 in N-bag
step 4: we put ball-4 in N-bag and another ball-4 in E-bag
etc.

Which bag has the most balls, N-bag or E-bag? In mathematical terms, has the set of natural numbers more elements than the set of even numbers? Intuition would say so, as we fill the N-bag at twice the pace of the E-bag. There is also this fantastic demonstration I have found.

DEMONSTRATION
Let's call N the number of elements of the N-bag, and E the number of elements of the E-bag. Let's build a bag made only with odd numbers, the O-bag. To make this bag, we take out from the N-bag an exact copy of the E-bag. Now, if the N-bag and the E-bag would have the same number of elements, the O-bag would be empty. However, by construction, the O-bag is not empty, therefore the N-bag has more elements than the E-bag.

In more mathematical language, if we call N the number of elements of the Natural number set, and E the number of elements of the even number set, then O, defined as the number of elements of the odd number set is O=N-E.
If N=E than O=0.
But there is a number 3 which is odd, and therefore O cannot be 0, which means N is not equal to E.



It looks great, only it is wrong! N=E=O, but the demonstration does not fit the margin of this post.