Cantor's first set theory article: Difference between revisions

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! style="background: f5f5f5;" |'''Proof that the number generated is {{sqrt|2}} − 1'''

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| style="padding-left: 1em; padding-right: 1em" |The proof uses [[Farey sequence]]s and [[simple continued fractionsfraction]]s. The Farey sequence <math>F_n</math> is the increasing sequence of [[completely reduced fraction]]s whose denominators are <math>\leq n.</math> If <math>\frac{a}{b}</math> and <math>\frac{c}{d}</math> are adjacent in a Farey sequence, the lowest denominator fraction between them is their [[mediant (mathematics)|mediant]] <math>\frac{a+c}{b+d}.</math> This mediant is adjacent to both <math>\frac{a}{b}</math> and <math>\frac{c}{d}</math> in the Farey sequence <math>F_{b+d}.</math><ref>{{harvnb|LeVeque|1956|pp=154&ndash;155}}.</ref>

Cantor's construction produces mediants because the rational numbers were sequenced by increasing denominator. The first interval in the table is <math>(\frac{1}{3}, \frac{1}{2}).</math> Since <math>\frac{1}{3}</math> and <math>\frac{1}{2}</math> are adjacent in <math>F_3,</math> their mediant <math>\frac{2}{5}</math> is the first fraction in the sequence between <math>\frac{1}{3}</math> and <math>\frac{1}{2}.</math> Hence, <math>\frac{1}{3} < \frac{2}{5} < \frac{1}{2}.</math> In this inequality, <math>\frac{1}{2}</math> has the smallest denominator, so the second fraction is the mediant of <math>\frac{2}{5}</math> and <math>\frac{1}{2},</math> which equals <math>\frac{3}{7}.</math> This implies: <math>\frac{1}{3} < \frac{2}{5} < \frac{3}{7} < \frac{1}{2}.</math> Therefore, the next interval is <math>(\frac{2}{5}, \frac{3}{7}).</math>