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TakuyaMurata (talk | contribs) →A proof using the contraction mapping principle: more precise version of the lemma (which is needed in the real closed field case) |
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for all <math>x, y</math> in <math>B(0, r)</math>, then for <math>f = I + g</math> on <math>B(0, r)</math>, we have
:<math>(1-c)|x - y| \le |f(x) - f(y)|,</math>
in particular, ''f'' is injective. If, moreover, <math>g(0) = 0</math>, then
:<math>B(0, (1-c)r) \subset f(B(0, r)) \subset B(0, (1+c)r)</math>. More generally, the statement remains true if <math>\mathbb{R}^n</math> is replaced by a Banach space. Also, the first part of the lemma is true for any normed space.}}
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