Title: Koch Fractal, the concept of Limit and Real-Analysis Post by: Finer on November 08, 2014, 11:52:39 PM 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: (http://farm5.staticflickr.com/4015/4430320710_686e9e991b.jpg) 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/4J)*4J, 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. |