Cauchy sequence exercise question (Exercise 2.6.4 Abbott analysis)

Question

I have a problem solving the question below. I'm stuck at the stage (ii) of my solution below.

Problem: Let $(a_n)$ be a Cauchy sequence. Decide whether the following sequence is a Cauchy sequence, justifying each conclusion.

$c_n = (-1)^n a_n$

My attempt: Since $|c_n - c_m| = |(-1)^n a_n - (-1)^m a_m|$, I need to consider parities of n and m.

i) if n, m are both even or odd, $|c_n - c_m| = |(-1)^n a_n - (-1)^m a_m| = |a_n - a_m|$. Since $(a_n)$ is a Cauchy sequence, $|c_n - c_m|$ is a Cauchy sequence when n and m are both even or odd.

ii) if n, m have opposite parity, then $|c_n - c_m| = |-c_n - c_m| = |c_n + c_m|$. (And I'm stuck from here).

Thus, I sense that this is not a Cauchy sequence and I want to show it, but how? I considered the negation of the definition of Cauchy, but I'm not sure how to proceed at this point.

Answer 1

it is Cauchy iff it converges. So if the limit of $a_n$ is zero then so is the limit of $c_n$. Otherwise there is no limit and so no Cauchy.

Answer 2

Let $a_n = 1$ $\forall n$, obviously $a_n$ converges to 1 and is a Cauchy sequence. Now $c_n=(-1)^n a_n$ just oscillate between +1 and -1 and cannot possibly be a Cauchy sequence. So your intuition (ii) is correct.

By the way, if you want to disprove a statement, you only need 1 counter example.

Answer 3

If $a_n$ is Cauchy then is convergent to some $l$. If $l\ne 0$ then the sequence $(-1)^na_n$ roughly say, oscillates between $-l$ and $l$ and is not convergent as well as not Cauchy. If $l=0$ then that secondary sequence is also Cauchy.