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Sequence and Cauchy sequence

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1)Find a sequence of real numbers {a_i} such that lim┬(n→∞) a_n does not exist , but it has 3 subsequential limits of 0,1,2.

2) Assume {p_n}, and {r_n} are Cauchy sequences in ℝ . Using the definition of a Cauchy sequence prove that {p_n+r_n} is a Cauchy sequence in ℝ.

3) Prove that {1/n^2 } is a Cauchy sequence in ℝ.

4) Assume {p_n}, and {r_n} are Cauchy sequences in ℝ . Using the definition of a Cauchy sequence prove that {p_n+r_n} is a Cauchy sequence in ℝ.

5). Suppose that {a_i} is a Cauchy sequence in a metric space X,d and lim┬(n→∞) a_n=p. Suppose, in addition, {b_i} is a sequence such that d(a_n,b_n)<1/n for all integers n. Prove that lim┬(n→∞) b_n=p.

6) 3. Let {a_i} be a sequence in ℝ. Prove lim┬(n→∞) a_n=0, if and only if, lim┬(n→∞) |a_n |=0.

If lim┬(n→∞) |a_n |=1, is it true that lim┬(n→∞) a_n=1? Prove your answer

7) Let {a_i} be a sequence in a metric space X,d. Prove that lim┬(n→∞) a_n=p if and only if lim┬(n→∞) d(a_n,p)=0.

8) Suppose that {a_i} is a sequence in ℝ such that lim┬(n→∞) a_n=0. Suppose that {b_i} is a sequence in ℝ such that |b_i |≤M, for all i where M is some positive real number. Prove from the definition of a limit of a sequence that

lim┬(n→∞) 〖(a〗_n b_n)=0.

Skills: Mathematics

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