40 2.1 Convergence 5n2 + n+ 1 5 (c) lim = . n→∞ 3n2 + 2n − 7 3 3n2 + 5 1 (d) lim = . n→∞ 6n2 + n 2 2n2 + n+ 5 2 (e) lim = . n→∞ 5n2 + 1 5 2n + 1 2 (f) lim = . n→∞ 3n − 1 3 2n3 + 1 1 (g) lim = . n→∞ 4n3 − n 2 2 − 4 1 n (h) lim = 4. n→∞ n2 − n 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. sinn 2.1.5 Prove that limn →∞ = 0. n 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.) 2 − 3) n+ cos(n (a) lim n→∞ 2n2 + 1 3n (b) lim n→∞ n! n! (c) lim n→∞ nn 2n (d) lim (Hint: see Exercise 1.3.3(d)). n→∞ 3n 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 A be a non-empty subset of real numbers bounded above and let α = supA. Prove that there is a sequence {xn} in A which converges to α. 2.1.11 Prove that if limn →∞ an = a > 0, then there exists N ∈ N such that an > 0 for all n ≥ N. an+1 2.1.12 ▶ Prove that if limn→∞ an = a ≠ 0, then limn →∞ = 1. Is the conclusion still true if an a = 0? 2.1.13 Let {an} be a sequence of real numbers such that limn →∞ an = 3. Use Defnition 2.1.1 to
RkJQdWJsaXNoZXIy NTc4NTAz