Let us return to the proof without words of S=(2a+2b+2c+2d+…) < X:
[qimg]http://farm5.static.flickr.com/4015/4430320710_daf5b36c0f_o.jpg[/qimg]
Some direct perception notions:
1) S or X are sizes > 0.
2) X is a constant size > 0, which is not changed by any amount of bends along it.
3) S is the result of the added projected sizes of the bended forms of X upon the non-bended form of X.
4) The added end-points or the infinitely many points that are located along the infinitely many bended versions of X, contribute exactly 0 size to S value, so S value is the result of the infinitely many convergent sizes (where each size > 0) 2a+2b+2c+2d+… such that for any 0 size (that does not contribute anything to the result) there is a size > 0, that does contribute to the result of S.
5) Since there are infinitely many convergent sizes > 0, then S < X exactly by 0.000…3/4, which is the invariant ratio among the infinitely many projected (and convergent) bended versions of X upon the non-bended version of X, where S is a fog exactly as the infinitely add points along the infinitely many bended versions of X, are a sum (=0).
(Please pay attention that this direct perception is truth for any absolute value of X or S.)
6) (5) can't be understood by a cyclopean view local-only reasoning (see The Man's case).