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] Inhalt 1 Formale Aussage 2 Nachweisen 3 Anmerkungen 4 References Formal statement If {Anzeigestil X} is a compact topological space, und {Anzeigestil (f_{n})_{nin mathbb {N} }} is a monotonically increasing sequence (meaning {Anzeigestil f_{n}(x)leq f_{n+1}(x)} für alle {Anzeigestil nin mathbb {N} } und {Anzeigestil xin X} ) of continuous real-valued functions on {Anzeigestil X} which converges pointwise to a continuous function {displaystyle fcolon Xto mathbb {R} } , then the convergence is uniform. The same conclusion holds if {Anzeigestil (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. Für jeden {Anzeigestil nin mathbb {N} } , Lassen {Anzeigestil g_{n}=f-f_{n}} , und lass {Anzeigestil E_{n}} be the set of those {Anzeigestil xin X} so dass {Anzeigestil g_{n}(x)
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