Mathematical circles squared; a third collection of by Howard W. Eves

By Howard W. Eves

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G admet c comme limite en b donc : ∃d > 0 ∀y ∈ B 56 y−b F d ⇒ g(y) − c G ´. Rapport CCP, 2002 « Le langage est très souvent d’une rare imprécision : par exemple, sur une indétermination du type l’infini à la puissance 0, un candidat dira que la limite est 1 car “l’exposant l’emporte sur la limite” ». 2. Topologie, étude locale des applications Un tel d > 0 étant déterminé, on sait aussi que : ∃m > 0 ∀x ∈ A x −a E m⇒ f (x) − b d. F On en déduit : ∀x ∈ A x −a E m ⇒ g( f (x)) − c G ´. Pour s’entraîner : ex.

Récapitulons : ∀n ∈ N n sup(N1 , N2 ) ⇒ vn − l u n − vn + u n − l 2´. La suite v converge donc vers l. Théorème 28 Soit (u n ) une suite d’éléments de E convergeant vers l. Si l = 0 E , la suite (u n ) est équivalente à la suite constante (l) : u n ∼ l. Dans ce cas, il est essentiel que l soit non nulle. (Considérer la suite (1/n)). D’autre part, u n ∼ 0 E ne signifie pas que la limite de (u n ) soit nulle, mais que la suite (u n ) est identiquement nulle à partir d’un certain rang. 4. Cas des suites numériques Les suites numériques (ou scalaires) sont les suites de réels ou de complexes.

Montrons qu’elle a une unique valeur d’adhérence. La suite (an ) admet une valeur d’adhérence l, limite d’une suite 9 On a : f (a p ) = a p car a p est une valeur d’adhérence de u. La suite est alors stationnaire à partir du rang p, ce qui contredit V ad(u) = [c, d]. Donc V ad(u) = {l}. La suite (an ) converge donc vers l. Comparaison de suites Dans ce paragraphe, (u n ) est une suite d’éléments du K-espace vectoriel normé (E, E ) et (an ) une suite d’éléments de K. 1. Relation de domination La suite (u n ) est dite dominée par la suite (an ) si : ∃K ∈R ∃N ∈N ∀n N un K |an |.

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