MATH 401 INTRODUCTION TO ANALYSIS-I, SPRING TERM 2016, PROBLEMS 11 Return by Monday 4th April 1. Prove that if x > 0 and xn is a sequence with limn xn = x, then there is a real number N such that whenever n > N we have xn > 0. 2. Prove that 3n5 4n3 + 2n + 7 3 = . 5 + 5n4 + 6n3 + n2 + 1 n 4n 4 lim 3. Let an = (1)n . We have seen in class that an diverges. Dene bn = a1 + + an , n the \"average\" of an . Prove that bn converges. 4. Suppose that 0 < k < 1 and xn satises |xn+1 | < k|xn | for n = 1, 2, 3, . . . . Prove that (i) |xn | k n1 |x1 |, (ii) limn xn = 0