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Extra info for An Explicit Approach To Elementary Number Theory

Example text

Ein Beispiel einer Intervallschachtelung ist ([en, e~]) mit en: = (1 + ~)n und e~ : = (1 + ~l + 1. Diese Schachtelung bestimmt die nach L. EULER benannte reelle Zahl e = 2,71828 ... , die in der Analysis beim Logarithmus und der Exponentialfunktion grundlegend wichtig ist, siehe auch Kap. 5. Es ware nunmehr notig, fUr die Aquivalenzklassen der Schachtelungen eine Addition, eine Multiplikation und eine Anordnung zu definieren und nachzuweisen, daB die Axiome (R 1}-(R3), die zu Beginn des § 2 ausgesprochen wurden, erfiiIlt sind.

Da P kein Minimum hat, muB inf pea. sein. Da a < b fUr aIle a e a. , das heiBt, inf P ist das Maximum von a.. (b) - (e): Sei (b n) eine nach unten beschrankte, monoton fallende Folge. Man erhiilt dureh a. ,P). Naeh (b) hat a. ein Maximum s. Dann konvergiert (b n) nach s. Beweis dazu: Wenn Ii > 0 vorgegeben ist, existiert ein Index k, so daB bk < s + Ii ist. Denn wenn s + Ii ~ bk fUr aIle k ware, ware s + Ii e a. im Widersprueh zu s = max a.. Da (b n ) monoton faIlt, ist bm ~ bk fUr aIle m ~ k. AuBerdem ist s ~ bm fUr aIle m, also s ~ bm ~ bk < S + Ii fUr aIle m ~ k.

GemaB (2). ]) eine Intervallschachtelung. Wenn man § 5. Axiomatische Beschreibung der reellen Zahlen 39 anstatt von (an) von einer anderen Fundamentalfolge (a~) ausgeht, so daB (a~ - an) eine Nullfolge ist, und dazu (r~) und (s~) gemiiB (2) wiihlt, ist ([r~, s;,]) zu ([rno sn]) iiquivalent. Man hat also eine wohlbestimmte Abbildung von den Fundamentalfolgen modulo den Nullfolgen in die Aquivalenzklassen der Schachtelungen. Diese Abbildung ist zu der oben beschriebenen invers. Praktische Vorteile der Intervallschachtelung gegeniiber Schnitten oder Fundamentalfolgen sind folgende: Wenn (In) die reelle Zahl x beschreibt, wird durchjedes In die Lage von x auf der Zahlengeraden innerhalb bestimmter Schranken fixiert.

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