WE DEFINE a function F(x) which takes every real value in every interval.
Express the fractional part of x as a binary "decimal" (which may or may not terminate) 0.x1x2x3 ... .. .. .. (1)
We denote the sequence 0111 ... (n 1's)0 by Sn and define
| fn = 0 if Sn occurs a finite number of times in (1) | |
| fn = 1 if Sn occurs an infinite number of times | |
| and | f(x) = 0.f1f2f3 ... (another binary decimal). |
Now take any interval of x (as small as we like). We can find N.x1x2...xr such that with these fixed x is bound to lie in the interval. We can now make f(x) take any value 0.f1f2... between 0 and 1. For consider the sequence n1,n2,n3, ... (which may terminate) of all the n's for which fn = 1, and write
x = N.x1x2...xr; Sn1; Sn1, sn2; Sn1, Sn2, Sn3; ...
which contains all Sni an infinite number of times and all other Sn a finite number of times. By considering
we
have a function which takes every real value in every interval of
x.
Eureka, 20.
Reproduced from Eureka 27 page 31.
HTML conversion Copyright © 2002-4 The Archimedeans.
Errata: Sn1, Sn2, Sn3, Sni should read Sn1, Sn2, Sn3, Sni (multiple places).
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