By Michael Zakharyaschev

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**Example text**

1’ 2 -. kl n+l 1 n+2 --(n+l)! n + l n = sn+r,, . m(n-1)! = n+l (n+l)! n’ n! s,+n! rn . This is impossible, for n! r, < ljn is not an integer. L ~ A S S . This is a well-known proof for the irrationality of e , but does it settle the order relation between e and mjn? INT. To do that, we must convert the last part into a positive reasoning. We note that 1

D e f i n i t i o n . A number-generator of the form {zn2-"), where every x,, is an integer, and which satisfies ( 2 ) , will be called a canonical number-generator. We have proved : 42 SPREADS AND SPECIES T h e o r e m 1 . Every real number z coincides with a. canonical number-generator { 4 2 - " } which satisfies ( 1). It is clear from the proof, that in (1) the factor 518 can be replaced by 1 / 2 + ~ , , , where c,,,>Q. 4. 1. Continuity The notion of an ips is an extension of that of a sequence given by a law.

Point-generators and p i n t s Definition 1. ) of real number-generators. If &= (&I, . . , ttn, . ) (i = 1, 2 ) , then will often be identified with the ips ((611, Ezr), . . , ( t i n , &n), . -1. Definition 2 . A point x of the plane is an ordered pair (xl,x2) of real numbers. I leave it to the reader to supply the definitions of coincidence between two point-generators between two points and between a point-generator and a point. Theorem 1 . Every point-generator determines one and only one point with which it coincides.