Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >...

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m > 1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the sequence is an integer plus or minus 1/3 ). Show that the sequence sn is eventually constant, i.e. after a point all terms of the sequence are the same

Expert Solution

Colby Messinger answered 2 years ago

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all...

Let (sn) be a sequence that converges. (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a. (b) Show that if sn ≤ b for all but finitely many n, then lim sn ≤ b. (c) Conclude that if all but finitely many sn belong to [a,b], then lim sn belongs to [a, b].

Recall that a sequence an is Cauchy if, given ε > 0, there is an N...

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Let sn = 21/n+ n sin(nπ/2), n ∈ N. (a) List all subsequential limits of (sn). (b) Give a formula for nk such that (snk) is an unbounded increasing subsequence of (sn). (c) Give a formula for nk such that (snk) is a convergent subsequence of (sn).

Let S = (s1, s2, . . . , sn) be a given sequence of integer...

Let S = (s1, s2, . . . , sn) be a given sequence of integer numbers. The numbers can be positive or negative. We define a slice of S as a sub- sequence (si,si+1,...,sj) where 1 ≤ i < j ≤ n. The weight of a slice is defined as the sum of its elements. Provide efficient algorithms to answer each of the following questions: a)Is there any slice with zero weight ? b)Find the maximum weight slice in...

Consider the sequence sn defined as: s0 = 1 s1 = 1 sn = 2sn-1 +...

Consider the sequence sn defined as: s0 = 1 s1 = 1 sn = 2sn-1 + sn-2 What is the base case for this recursive relation? Find s5 Write the pseudocode for a recursive function to find Sn for any arbitrary value of n. Create a non-recursive formula for finding the nth term in the sequence in O(1) time.

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Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m >...

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