Electricite et optique by Poincare H.

By Poincare H.

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T h e procedure of completing the set % of rational numbers by embedding it in the set of all real numbers can be generalized. 6. in A? is everywhere dense in A. $ and 9 c+ 3. 3. 2. A'. T h e proof of this theorem can be given by generalizing Cantor's construction, by which one builds the set of real numbers from the rational numbers. &. } and &" = they are equivalent if and only i f lirn d(g,', n-ta 5,") = 0. 8) the general definition of an equivalence relation. 7. $ (3) transitive: [ - - 5 for all f E S; implies that 7 4; 5 implies that q and q 4 - 5.

2), and in the direction indicated on Fig. 2. A striking feature of the above experimental procedure is that it does not tell us, by itself, whether there was indeed a particle which had passed through the aperture while the shutter was open. This feature is common to many preparatory measurements. + We note that the above preparatory procedure cannot prepare a sharp value p of the momentum, but rather, due to the finite size of the apertures 0, , 0, , and 0, , it prepares a whole range A C 88, of momenta.

3) defines vector f' + 2 = J" operations on d,' which thus becomes a vector space. 4) . 32 I. 4) exists because (fiI gl), (fiI g2),... is a Cauchy sequence of numbers, as can be seen from the following inequality: I(fm namely, I gm) II~, II + - (fn I gn)l I(fm - f n I gwJ + (fn I Ern - gn)l < llfm - f n II II gm II 4-llfn II II gm - gn II; = W,0) = IIJII, and IIg, II - 112 IIR for fi -+ =, where while Ilf, & -fn 11 and 1) g , - g, )I can be made arbitrarily small for sufficiently large m and n.

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