## Calculus: A modern, rigorous approach by Horst R. Beyer

By Horst R. Beyer

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Proof. For this, let x1 , x2 , . . be a sequence of real numbers converging to some x È R and ε 0. Then there is n0 È N¦ such that xn ¡ x for all n È N¦ satisfying n xm ¡ xn εß2 n0 . The last implies that xm ¡ x ¡ Ôxn ¡ xÕ for all n, m È N¦ satisfying n Cauchy sequence. xm ¡ x n0 and m xn ¡ x ε n0 . Hence x1 , x2 , . . is a That every Cauchy sequence of real numbers is convergent is a deep property of the real number system. 11 in the framework of Cantor’s (1872) construction of the real number system by completion of the rational numbers using Cauchy sequences.

5 Fig. 23. 8 1 Fig. 27. 59 x where a and b are real numbers such that a such that f Ôx0 Õ f ÔxÕ Ô f Ôx0 Õ for all x È Öa, b×. b. Then there is x0 È Öa, b× f ÔxÕ Õ Proof. For this, in a first step, we show that f is bounded and hence that sup f ÔÖa, b×Õ exists. In the final step, we show that there is c È Öa, b× such that f ÔcÕ sup f ÔÖa, b×Õ. For this, we use the Bolzano-Weierstrass theorem. The proof that f is bounded is indirect. Assume on the contrary that f is unbounded. Then there is a sequence x1 , x2 , .

13. See Fig 15. 15. Let S be a non-empty subset of R. We say that S is bounded from above (bounded from below) if there is M È R such that x M (x M) for all x È S. 16. Let S be a non-empty subset of R which is bounded from above (bounded from below). Then there is a least upper bound (largest lower bound) of S which will be called the supremum of S (infimum of S) and denoted by sup S (inf S). Proof. First, we consider the case that S is bounded from above. For this, we define the subsets A, B of R as all real numbers that are no upper bounds of S and containing all upper bounds of S, respectively, A: B: Øa È R : There is x È S such that x Øb È R : x b for all x È S Ù .