![]() (It is well known that A = C iff X is compact. Real functions on X, B the set of real functions sending C-sequences intoĬ-sequences and C the set of continuous real functions. Subject: Re: Re: Re: Uniform Continuity and Cauchy Sequences So f(x_n) -> f(x) as well, and f is continuous. ![]() 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 between uniform convergence on Cauchy sequences and uniform e. >sends every Cauchy sequence of (X,d) to a Cauchy sequence of converges uniformly to zero on each Cauchy sequence E of the reals. 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 Indeed, a Cauchy sequence in X is convergent => its image is alsoĬonvergent by continuity => the image is Cauchy. 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
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |