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Problem 2 (2.12.2) Show directly that any multiple of a Cauchy sequence is again a Cauchy sequence. Proof. Let {sn} be a Cauchy sequence of real numbers and let ce R. Let e > 0. If c = 0, then CSn = 0 for all n e N so that \csn csm= |0 - 0 = 0 N since {sn} is a Cauchy sequence. Therefore, if n, m > N, we have |csn csm/ = |el|sn sml 038 > 0(0 0, there exists a 8 >0 such that 0 0. If c = 0, then CSn = 0 for all n e N so that \csn csm= |0 - 0 = 0 N since {sn} is a Cauchy sequence. Therefore, if n, m > N, we have |csn csm/ = |el|sn sml 038 > 0(0 0, there exists a 8 >0 such that 0