Does anyone have a concrete example of a Cauchy sequence that is not convergent?
It all depends on which spaces you're working. For example in working with $\displaystyle \mathbb{R}$ we have that $\displaystyle x_n=\left( 1+\frac{1}{n} \right) ^n$ is convergent, but if you're in $\displaystyle \mathbb{Q}$ this sequence is Cauchy but not convergent.