- 27 add comment up vote 1 down vote For case a), we want to show that for any given a, and every open set containing a, that all the an are in the open set as long as n is greater than some M which we get to choose. Since the open set in only missing a finite number of elements, we can put them in order {b1,b2,…,bp}. Then in the seriesan, since b1 does not appear infinitely many times, there is a last occurrence, call it b1f. Similarly there is a last occurrence of b2, call it b2f. Then if n>M=max(b1f,b2f,…bpf), an will be in our open set. So the sequence converges to a The arguments for b) and c) are similar. shareimprove this answer answered Nov 6 '11 at 19:48 Ross Millikan 118k973144 add comment Your Answer Sign up or log in Sign up using Google Sign up using Facebook Sign up using Stack Exchange Post as a guest Name Email By posting your answer, you agree to the privacy policy and terms of service. Not the answer you're looking for? Browse other questions tagged general-topology convergence or ask your own question. asked 2 years ago viewed 804 times active 5 months ago Get the weekly newsletter! Top questions and answers Important announcements Unanswered questions see an example newsletter Related 10 Is there a way of working with the Zariski topology in terms of convergence/limits? 9 Convergence in topologies 2 In a first-countable space, an accumulation point of the set of terms in a sequence is also a limit-point of the sequence 5 Prove that C[X] with the pointwise convergence topology is not metrizable 7 Convergence in the Box Topology. 1 Convergence of a sequence in a nested local base -1 Let X be any topological soace which satisfies the first axiom of countability. If every compact subset of X is closed, then X is Hausdorff. 0 Convergence of sequence in uniform and box topologies 0 Finding the limit of a converging sequence? 3 Convergent sequence in product space on ℝω Hot Network Questions Problem with a simple exercise Is 3d6 the same as 1d18? How can I protect a file from user changes? forest trees: rounded rectangle nodes overlap Is the following endgame position a draw or white has advantage? Batch Renaming Automation *_2.* TO *.* Difference between these two sentences python vs bc in evaluating 6^6^6 What can I do about my domains history? Useful strategies for answering "dumb" questions in a talk? How to deal with multiple enumeration types in one variable How to draw big circle

*American Mathematical Society ; Mathematics Advanced Study Semesters,**Providence, R.I.; Philadelphia,*(*2008*)- (
*2007*)