Let (In), and (yn).m-1 be sequences such that Pr – yn| < 1/n for all n. Use the definition of convergence to prove that, if (2n)_1 is convergent, then (Yn)-1 is convergent.
(5) Use induction to show that Ig(n) <n for all n > 1.
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
(1) Does there exist a sequence of 10100 consecutive composite numbers? (2) Let Pn be the nth prime number. Show that pn < 22".
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
1. Let L = {ambm cn | m <n}. Use the pumping lemma to show that L is not a CFL.
Given that the sequence defined by - 1 2+1 = 5-1 an is increasing and an < 5 for all n. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.)
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
Solve the inequality. 2x + 3 -1< 5