By Szymon Dolecki

ISBN-10: 2705687416

ISBN-13: 9782705687410

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Extra resources for Analyse fondamentale : espaces métriques, topologiques et normés

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Puisque x = limn_>00a;n, il existe n i tel que x n G Bi(x) pour tout n > ni, et comme xn = limfc_KX):rn}fc, il existe f l : Nni -> N telle que x n>k G B i(x) pour tout k > fi(n) et n > ni. Si nous avons trouvé des nombres naturels n\ < ri2 < ... < np et fl, /2. • • •, fp tels que f q : N9 -> N pour 1 < q < p avec f q(n) < f q+i(n) pour n G N9+ x, et n > n q, k > f q(n) = > a G Bi(x), Q alors, il existe np+i > rip et f p : Np -> N avec f p(n) < f p+i(n) telle que x Htk G Bi ( x) pour tout n > n p et k > f p(n).

Dans un espace topologique, une suite peut converger vers plusieurs points. C’est pourquoi la notation limn_>00xn adoptée dans des espaces métriques et dénotant la limite de (xn)n, n’est pas adéquate. On note Limn_>oo x n la limite de (xn)n, c’est-à-dire l’ensemble des éléments vers lesquels (xn)n converge. Par conséquent, une suite (x n)n est convergente si et seulement si Limn-^oo x n ÿé 0 . Si Limn_>0o %n est un singleton, alors on désigne lim ^oo x n son élément. Si £ > r, alors Lim ^oo x n C Lim ^oo xn, car V^(x) a plus d’éléments que VT(x).

3. Si f , g : X —>Y sont deux applications continues et f \ A = 9\a >° ù A est une partie dense de X , alors f = g. D é m o n s t r a t i o n . Comme A est dense, pour tout x e X il existe une suite (xn)n dans A telle que x = lim^ooXn. D’après la continuité de / et 9, on a f(x) = limn_>oo f ( x n) = limn_>oo g(xn) = g(x). □ Une application bijective continue / : X —» Y s’appelle un homéomor­ phisme si f ~ l est également continue. 4. Soit X ~ {£ : n G N} U {0} C IR. 2 et d(x,y) = \ x - y \ (héritée de la droite réelle M).

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Analyse fondamentale : espaces métriques, topologiques et normés by Szymon Dolecki

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