1) (a) Determine if the following statements are true or false. If true give a reason or cite a theorem and if false, give a counterexample.

Question

i) If {an} is bounded, then it converges.

ii) If {an} is not bounded, then it diverges.

iii) If {an} diverges, then it is not bounded.

(b) Give an example of divergent sequences {an} and {bn} such that {an+bn} converges.

Answer 1

(i) is FALSE. For example, {a_n} = {1, -1, 1, -1, ...} is bounded, but diverges.

(iii) is the contrapositive of (i), so is also FALSE. You can use the same counterexample as in (i).

(ii) is TRUE

(b) Let (a_n} = (1, -1, 1, -1,...) and {b_n} = {-1, 1, -1, 1, ...) . {a_n} and {b_n} both diverge, but

{a_n+b_n} = {0, 0, 0, ...} converges.