3. Let 2n2 + 3 |an| < for alln E N. n3 + 5n2 + 3n +1 Show (using definition of Cauchy Sequence) that {an} is a Cauchy sequence.

(d) Consider the sequence {an}CQ given by п 3 2,4, 7, 17 n+1 2n otherwise Prove the sequence is Cauchy from the definition of a Cauchy sequence.

a) Suppose (an) is Cauchy and that for every k∈N, the interval (−1/k,1/k) contains atleast one term of (an). Can we say that (an) converges to 0? Either show that it does or give a counter-example. b) Suppose (xn) is Cauchy and (yn) is bounded and monotone. Can we say anything aboutwhether the sequence (xn yn) converges or not? Justify your answer!

Exercise 2 Use the formal definition of a Cauchy sequence to show that the sequence (an) sequence. +31 = 4n+1 √n=1 is a Cauchy