Prove that each of the following statements is false by giving a counter example

Question

Prove that each of the following statements is false by giving a counter example. a) if a sequence {an} is monotonic, then it is convergent.
b) if both sequences {an} and {bn} are convergent, then {an/bn} is convergent.
c) if sequence {an} is monotonic then it is divergent. Prove that each of the following statements is false by giving a counter example. a) if a sequence {an} is monotonic, then it is convergent.
b) if both sequences {an} and {bn} are convergent, then {an/bn} is convergent.
c) if sequence {an} is monotonic then it is divergent. a) if a sequence {an} is monotonic, then it is convergent.
b) if both sequences {an} and {bn} are convergent, then {an/bn} is convergent.
c) if sequence {an} is monotonic then it is divergent.

Explanation / Answer

a)

If a sequence {an} is monotonic, then it is convergent. : False

Counter example : {an} = {1,2,3,4,5,6,.......}

This sequence is monotonic but not convergent as at n->infinity , an->infinity .

b)

If both sequences {an} and {bn} are convergent, then {an/bn} is convergent :False

Counter example : {an} = 1 and {bn} = 1/n

{an} and {bn} are both convergent.

Here , {an/bn} = n which approaches infinity as n->infinity. Hence not convergent.

c)

If sequence {an} is monotonic then it is divergent. : False

Counter Example : {an} = {1/n}

This sequence is monotonic but not divergent as at n->infinity , an-> 0.

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