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Unformatted text preview: sions s = 0 .d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 . . . with digits satisfying d i d i +1 for all i . For example, . 97777665555555555555555555555555 . .. ending in innitely many 5s is in S , but / 10 = 0 . 314159265358979323846 . . . is not in S . III. Suppose E R is a compact set with a countable (innite) number of elements. For each question below, provide an example or prove the impossibility. (i) Can the set of limit points of E in R be empty? (ii) Can the set of limit points of E in R have exactly 5 elements? (iii) Can the set of limit points of E in R contain the set Q (0 , 1) ? (iv) Can the set of limit points of E in R contain all the integers Z ?...
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