By Bourbaki N.

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Since was arbitrary, 1 + UF claimed. 2 (The Archimedean Principle for Real Numbers) If : and ; are positive real numbers, then there is some positive integer n such that n: ;. Proof. The proof will be by contradiction. Suppose that there exist positive real numbers : and ; such that n: n ; for every natural number n. Since : 0, : 2: 3: n: is an increasing sequence of real numbers that is bounded above by ;. Since U n satis¿es the least upper bound property n: : n + Q has a least upper bound in U, say L.

Since : 0, : 2: 3: n: is an increasing sequence of real numbers that is bounded above by ;. Since U n satis¿es the least upper bound property n: : n + Q has a least upper bound in U, say L. Choose > 12 : which is positive because : 0. 1, there exists s + n: : n + Q such that L s n L. If s N :, then for all natural numbers m N , we also have that L m: n L. Hence, for m N , 0 n L m: . In particular, 0 n L N 1: 1 : 2 0 n L N 2: 1 : 2 and 1 Thus, L 12 : N 1: and N 2: L L 2 :.

1. the inverse of R, denoted by R 1 , is y x : x y + R 2. 5 For R x y + Q ] : x 2 y 2 n 4 and S x y + U U : y 2x} 1, R 1 0 1 1 1 1 1 0 2, S 1 | x 1 , and S i R 1 1 1 3 1 1 2 1. x y + U U : y 2 Note that the inverse of a relation from a set A to a set B is always a relation from B to A this is because a relation is an arbitrary subset of a Cartesian product that neither restricts nor requires any extent to which elements of A or B must be used.