Unformatted text preview: L . (c) (8 marks) Let x 1 ∈ R and deﬁne a sequence of real numbers { x n } by x n +1 = x 2 n + x n + 1 , n ≥ 1 . Show that the sequence { x n } does not converge. (3) (22 marks) Let A ⊂ R . (a) (3 marks) Explain what it means to say that x ∈ R is a cluster point (a.k.a. limitpoint; accumulation point) of A . (b) (3 marks) Explain what it means to say that the set A is bounded . (c) Now let S = { x ∈ R : x is a cluster point of A } . (i) (8 marks) Show that S is a closed set. (ii) (8 marks) Suppose A is bounded. Show that S is a compact set. [END OF PAPER]...
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 '08
 PROTSAK
 Math, Topology, Metric space, Closed set, Xn

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