Disproving that if $x^2_n$ is Cauchy, then $x_n$ is Cauchy

Question

I tried to work backwards from the fact that $x^2_n$ is cauchy to find a case where $x_n$ is not cauchy.

$\vert x^2_n-x^2_m \vert \lt \epsilon$

$\vert (x_n-x_m)(x_n+x_m) \vert \lt \epsilon$

$\vert x_n-x_m \vert \vert x_n+x_m \vert \geq \vert (x_n-x_m)(x_n+x_m) \vert \lt \epsilon$ (proved earlier in class)

Then a case would exist that $\vert x_n-x_m \vert \gt \epsilon/(\vert x_n+x_m\vert)$, making $x_n$ not a Cauchy sequence since $\epsilon/(\vert x_n+x_m \vert)\gt0$

Is this a legitimate approach, or are there assumptions that aren't allowed? Just looking for feedback as this is for a graded assignment that I can only seek guidance, not solutions for.

Answer 1

You cannot have variable pairing with $\epsilon$.

A counterexample is the sequence defined by $(1,-1,1,-1,...)$, the absolute value is the constant sequence.