For each of the following statements, determine whether it is true or false and

Question

For each of the following statements, determine whether it is true or false and justify your answer. If it is false, give a (simple) counterexample.
a. If the sequence {an2} converges, then the sequence {an} converges.

b. If the sequence {an+bn} converges, then the sequences {an} and {bn} also converge.

c. If the sequences {an+bn} and {an} converge, then the sequence {bn} also converges.

d. If the sequence {|an|} converges, then the sequence {an} also converges.

So I know a, b, and d are false, but I don't understand convergence of sequences and don't know what would prove as counterexamples and I need help proving c.

Explanation / Answer

a) False

counterexample: an = (-1)n is divergent but a2n = (-1)2n = 1 is convergent.

b) False

an = 2n , bn = -2n are divergent but an+bn = 0 is convergent.

c) True

bn = an+bn - an , thus if the sequences {an+bn} and {an} converge, then the sequence {bn} also converges

d) False

counterexample: an = (-1)n is divergent but |an| = 1 is convergent.

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