Prove $\{(-2)^n\}$ is not Cauchy by using the definition

Question

I've gotten that $|x_m - x_n|$ = 0 if $n=m$. I can't find the pattern to explain when $n+m$ is odd.

Answer 1

It's enough to find $n, n+1$ so that $x_n - x_{n+1}$ is not arbitrarily small even if picking $n$ large enough.

But $|(-2)^n - (-2)^{n+1}| = |2^n (1 - (-2))| = 3 \cdot 2^n$, so that for $\varepsilon < 3$, you cannot make the difference small enough by changing $n$ (it will always be $\geq 3$).