Homework Statement http://books.google.com/books?id=thAHAGyV2MQC&pg=PA20&lpg=PA20&dq=%22has+a+convergent+subsequence%22+topology+proof+compact&source=web&ots=yfNwPI7lv_&sig=6MWgB7WBS3P8NZ_a_7Dq_um7EvI Can someone explain there rational for the statement "for only finitely many indices j" in the second paragraph of the proof? Homework Equations The Attempt at a Solution