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In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence, in the sense that the convergence is uniform over the domain. (It is well known that A = C iff X is compact.Mode of convergence of a function sequence Real functions on X, B the set of real functions sending C-sequences intoĬ-sequences and C the set of continuous real functions. Since i Mi converges, the partial sums form a Cauchy sequence by Exercise 2.1. From this, and 3) (note that 3) is saying that the sequence (fn) converges uniformly to 0 ), it will follow that the series is uniformly Cauchy on D and thus uniformly convergent on D. However in your case since you are using sup norm then Cauchy sequences will converge uniformly. Essentially, such a series is alternating: for each x D, the series n 1( 1)n + 1fn(x) is a convergent alternating series. Then you will see that the definition of complete spaces does not require for Cauchy sequences to converge uniformly. Subject: Re: Re: Re: Uniform Continuity and Cauchy Sequences First you need to crasp what it means that a sequence converges uniformly. The limit function of a uniformly convergent sequence of continuous functions is. Meta ( Main Idea) Pointwise convergence is again a direct application of our basic knowledge regarding sequences of real numbers. A sequence is convergent if and only i f it is a Cauchy sequence. So f(x_n) -> f(x) as well, and f is continuous. The functions f n ( x) x n ( 1 x) converge both pointwise and uniformly to the zero function as n. Is a subsequence of it, this must be the limit. It converges (using completeness of (Y,d')), and as the constant sequence f(x) Then the sequence y_n = x_0, x, x_1, x, x_2, x.Īlso converges to x, hence is Cauchy, so f(y_n) is Cauchy in Y, so there Yes, this last example was the same that I had in mind.Īlso true: if (Y,d') is complete, then f is continuous (from this assumption).įor let x_n be any sequence converging to x. What IS true:If f is a uniformly continuous mapping ofĪ metic space X into a metric space Y,then in X. function on a complete metric space is a counterexample. >convergent by continuity => the image is Cauchy. Indeed, a Cauchy sequence in X is convergent => its image is also >sends every Cauchy sequence of (X,d) to a Cauchy sequence of Observe that if X is complete then a continuous function >In reply to "Uniform Continuity and Cauchy Sequences", posted by Dev on Feb 13, 2003: In reply to "Re: Uniform Continuity and Cauchy Sequences", posted by J. Subject: Re: Re: Uniform Continuity and Cauchy Sequences is, any sequence fn in B has a uniformly convergent subsequence. Indeed, a Cauchy sequence in X is convergent => its image is alsoĬonvergent by continuity => the image is Cauchy. (Pointwise convergence & Uniform Convergence). Sends every Cauchy sequence of (X,d) to a Cauchy sequence of >(X,d') sends every Cauchy sequence of (X,d) to a Cauchy sequence of >Suppose that a mapping f from a metric space (X,d) to a metric space In reply to "Uniform Continuity and Cauchy Sequences", posted by Dev on Feb 13, 2003: Subject: Re: Uniform Continuity and Cauchy Sequences (X,d') sends every Cauchy sequence of (X,d) to a Cauchy sequence of Suppose that a mapping f from a metric space (X,d) to a metric space Subject: Uniform Continuity and Cauchy Sequences Re: Re: Re: Uniform Continuity and Cauchy Sequences by J.Gerlits (February 14, 2003).Re: Re: Uniform Continuity and Cauchy Sequences by Henno Brandsma (February 14, 2003).Re: Re: Uniform Continuity and Cauchy Sequences by Al-Gadid (February 13, 2003).Re: Uniform Continuity and Cauchy Sequences by J.Uniform Continuity and Cauchy Sequences by Dev (February 13, 2003).
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