Koch Fractal, the concept of Limit and Real-Analysis

[Finer]

According to Real-Analysis a given line segment (some finite length > 0) has |R| points (where each point has exactly 0 length) along it.
  
Let's analyze the following diagram:



According to this diagram the series 2*(a+b+c+d+...) is the result of the projections of the endpoints of |N| folded line segments with invariant finite length X>0 upon itself, where each folded line segment is some degree of Koch Fractal.

There are |N| folded line segments because the cardinality of the sequence <4¹,4²,4³,...> (which is the number of sub-segments at each degree of Koch Fractal) is |N|.

As mentioned above, each one of the projected |N| degrees of Koch Fractal has an invariant finite length X>0 according to the general formula (X/K)*K (where K is the number of sub-segments along X for each degree, (in the case of Koch Fractal the formula is (X/4^J)*4^J, where j=1 to |N|)).  

In other words, 2*(a+b+c+d+...) (where the convergent sequence <a,b,c,d,...> has only |N| values) is < X if the invariant length X is observed from |R| points along it, because |N|<|R|.

So, the concept of limit holds (2*(a+b+c+d+...)=X) only if X is observed from |N| points along it.