40 2.1 Convergence (c) lim n→∞ 5n2 +n+1 3n2 +2n−7 = 5 3 . (d) lim n→∞ 3n2 +5 6n2 +n = 1 2 . (e) lim n→∞ 2n2 +n+5 5n2 +1 = 2 5 . (f) lim n→∞ 2n+1 3n−1 = 2 3 . (g) lim n→∞ 2n3 +1 4n3 −n = 1 2 . (h) lim n→∞ 4n2 −1 n2 −n =4. 2.1.3 Prove that if {an}converges to a, a∈R, then {|an|}converges to |a|. Is the converse true? 2.1.4 Let {an} be a sequence. Prove that if the sequence {|an|} converges to 0, then {an} also converges to 0. 2.1.5 Prove that limn→∞ sinn n =0. 2.1.6 Let {xn}be a bounded sequence and let {yn}be a sequence that converges to 0. Prove that the sequence {xnyn}converges to 0. 2.1.7 Prove that the following limits are 0. (Hint: use Theorem 2.1.3.) (a) lim n→∞ n+cos(n2 −3) 2n2 +1 (b) lim n→∞ 3n n! (c) lim n→∞ n! nn (d) lim n→∞ n2 3n (Hint: see Exercise 1.3.3(d)). 2.1.8 Prove that for every real number x there is a sequence of rational numbers {rn} which converges to x. (Hint: use the density property of Q, Theorem 1.6.3.) 2.1.9 Prove that for every real number x there is a sequence of irrational numbers {sn} which converges to x. (Hint: use the density of the irrational numbers, Theorem 1.6.5.) 2.1.10 Let Abe a non-empty subset of real numbers bounded above and let α=supA. Prove that there is a sequence {xn}inAwhich converges to α. 2.1.11 Prove that if limn→∞an =a>0, then there exists N∈Nsuch that an >0 for all n≥N. 2.1.12 ▶Prove that if limn→∞an =a̸ =0, then limn→∞ an+1 an =1. Is the conclusion still true if a=0? 2.1.13 Let {an} be a sequence of real numbers such that limn→∞an =3. Use Definition 2.1.1 to
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