Unformatted text preview: Theorem. (b) Show that the union of ﬁnitely many compact sets is compact. Problem 4. Prove the ∑ ∞ n =1 a n is convergent if and only if for any m ∈ N ∑ ∞ n =1 a m + n is convergent. Moreover, ∑ ∞ n =1 a n converges to a 1 + ··· + a m + ∑ ∞ n =1 a n + m . Problem 5. (a) State the Monotone Converge Theorem. (b) Deﬁne a 1 = 1 and a n = 31 a n1 for n ≥ 2. Determine the convergence or divergence of the sequence ( a n ). 1...
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 Spring '08
 STAFF
 Math, Topology, Compact space, Dominated convergence theorem, Algebraic Limit Theorem

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