Stating the behaviour of the sequence

Question

Assignment:

Q1. Give an example of a sequence {a_n} satisfying all of the following:

{a_n} is monotonic

0 < a_n < 1 for all n and no two terms are equal

lim_n→∞ a_nn = 1/2

Q2. Let k > 0 be a constant and consider the important sequence {kⁿ}.   It's behaviour as  n → ∞  will depend on the value of k.

(i) State the behaviour of the sequence as  n → ∞  when k = 1 and when k = 0.

(ii) Prove that  if k > 1 then kⁿ → ∞  as  n → ∞

(Hint: let k = 1 + t where t > 0 and use the fact that (1 + t)ⁿ > 1 + nt.

(iii) Prove that  if 0 < k < 1 then kⁿ → 0  as  n → ∞.

Provide complete and step by step solution for the question and show calculations and use formulas.