Dini's theorem

Dini's theorem In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform.[1] Contenu 1 Déclaration formelle 2 Preuve 3 Remarques 4 References Formal statement If {style d'affichage X} is a compact topological space, et {style d'affichage (F_{n})_{nin mathbb {N} }} is a monotonically increasing sequence (meaning {style d'affichage f_{n}(X)leq f_{n+1}(X)} pour tous {style d'affichage nin mathbb {N} } et {style d'affichage xin X} ) of continuous real-valued functions on {style d'affichage X} which converges pointwise to a continuous function {displaystyle fcolon Xto mathbb {R} } , then the convergence is uniform. The same conclusion holds if {style d'affichage (F_{n})_{nin mathbb {N} }} is monotonically decreasing instead of increasing. The theorem is named after Ulisse Dini.[2] This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.

Proof Let {displaystyle varepsilon >0} be given. For each {style d'affichage nin mathbb {N} } , laisser {style d'affichage g_{n}=f-f_{n}} , et laissez {style d'affichage E_{n}} be the set of those {style d'affichage xin X} tel que {style d'affichage g_{n}(X)N} et {style d'affichage x} is a point in {style d'affichage X} , alors {style d'affichage |F(X)-F_{n}(X)|

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