If $\lvert x_n \lvert$ is a sequence that satisfies $\lvert x_n \lvert \le \frac{2n^2+3}{n^3+5n^2+3n+1}$, then $\{ x_n \}$ is Cauchy.
I think that $\frac{2n^2+3}{n^3+5n^2+3n+1}$ is increasing with the lower bound of $\frac{1}{2}$. However, what if $x_n = \{\frac{1}{2}, -\frac{1}{2}, ...\}$, an alternating sequence satisfying this condition?
Can please someone point out where's my error?
First, you want to take the limit of this sequence. You can do that using l'Hopital's rule, or you could try multiplying by $$\frac{n^{-3}}{n^{-3}}.$$ If you do that, you'll notice many of the terms get small in the limit, and you can easily find the limit. (It's not $1/2$, actually.)
If the limit is zero, then the claim is true by the sandwich theorem. Otherwise, the counterexample you described (or a modified version) will work!