University of Illinois at Urbana Champaign Fall

University of Illinois at Urbana-Champaign Fall 2006 Math 444 Group E13 Graded Homework IX Due Monday, November 13. 1. Let (un) be a sequence of real numbers. We say that a ? R is an accumulation point of (un) if there exists a subsequence of (un) which converges to a. (a) What are the accumulation points of a convergent sequence ? (b) What are the accumulation points of the sequence un = cos(n pi 3 ) ? (c) Let (un) be a bounded, divergent sequence. Prove that it has at least two (distinct) accumulation points (Hint : why does there exist one accumulation point ? Can you use the fact that this point is not the limit of (un) ?) Correction. (a) If a sequence is convergent to a limit l, then all its subsequences are convergent to that same limit, so a convergent sequence has exactly one accumulation point : its limit. (b) One can see that for all n ? N one has u6n = 1, u6n+1 = cos( pi 3 ) = 1 2 , u6n+2 = cos( 2pi 3 ) = ? 1 2 , u6n+3 = cos(pi) = ?1, u6n+4 = cos( 4pi 3 ) = ? 1 2 , and u6n+5 = cos( 5pi 3 ) = 1 2 .

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