I saw this in some book once and it has been bugging me. The book had, I think as the first exercise it mentioned, to prove that the imaginary unit $i = \sqrt{-1}$ is not a limit of any real valued Cauchy sequence.
Given all I know of complex numbers, this shouldn't seem so hard to show. Like, I know any Cauchy sequence converges in $\mathbb R$ and elements of $\mathbb R$ have total ordering and elements of $\mathbb C$ cannot and so on, but still I have great doubts of how this is supposed to be done.
Hint: If $a_n$ is real, then the distance from $a_n$ to $i$ is $\sqrt{a_n^2+1}$, which is always $\ge 1$.
For an $\epsilon$-$N$ argument that the limit of the $a_n$ is not $i$, choose $\epsilon=\frac{1}{2}$.