Analysis, Manifolds and Physics. Basics by Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

By Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M.

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OZn+1 ... Zn+k (with at least n zeros after the decimal point). Each infinite decimal number is a real number. Proof. 00 .. OZn+1 ... • Each decimal number can uniquely be identified with an infinite decimal number. Proof. ZlZ2 ... Zm be a decimal number with exactly m decimal places. The sequence of numbers [all, [ah, ... , [a]n, ... ZlZ2 ... ZlZ2 ... zmO . 0 Z if n ::; m ifn> m (with n-m zeros after Zm in the second case) clearly is an infinite decimal number. • "Par abus de langage" we indicate the infinite decimal number produced by this proof by the same letter a as the decimal number itself.

Y > p. e. • Localization lemma. For any integer n > 1, any real numbers ao, a I, ... , an with ao < a 1 < ... < an, and any real number P with ao < P < an there exists a positive integer m < n such that am-I < P < am+l. Proof. The dichotomy lemma implies from al < a2 that one of the two inequalities P < a2 or a 1 < P must hold. In the case P < a2 (which is for sure if n = 2) 38 2. Real numbers we define m = 1 and thereby finish the proof. In the case n > 2 and a 1 < fJ we transfer the argument to the sequence ai, ...

Symmetry of the metric. The metric is symmetric: we always have IIx - y II = lIy - x II . Proof. This follows from a. , and from IIx - x II = 0: Taking z = y in the triangle inequality liz - xII - liz - yll :::; IIx - yll one gets Taking z lIy - x II :::; IIx - y II =x in the triangle inequality IIz-YII-lIz-xll:::; lIy-xll one gets • IIx - y II :::; lIy - x II . The symmetry of the metric allows us to formulate the triangle inequality as IIx -zll-IIY -zll :::; Ilx- yll or IIx -zll-liz - yll :::; Ilx - yll .

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