The Man said:
No Doron “S” is just a sum (in this case the sum of a convergent infinite series) and has nothing to do with your “inaccurate” fantasies.
No The Man “S” is just a fog (in this case the fog of a convergent infinite series) and has nothing to do with your “accurate” fantasies.
Look at this:
S=(0.9+0.09+0.09+0.009+...[base 10]) and it is a fog < 1 by fog 0.000...1[base 10]
As you see, all infinitely many added sums of the form above are not resulted by a sum, but they are resulted by a fog.
Exactly the same result is truth for all infinitely many finite bended Koch’s fractal forms that have constant sum X AND different endpoints, where S=(2a+2b+2c+2d+...) is exactly the projection of all these different endpoints upon the non-bended constant sum X.
Since all projections are the result of all bended Koch’s fractal forms that have constant sum X AND different endpoints, then the two different points are an invariant property of this projection, exactly because they belong to all infinitely many finite bended Koch’s fractal forms that have constant sum X,
that can’t be reduced to sum 0.
As a result the fog of S < the constant sum X.
This result is clearly seen by this proof without words: