Show $\displaystyle i^n/n$ is a cauchy sequence. need $\displaystyle |(mi^n-ni^m)/nm| < e $whenever $\displaystyle n,m>_N $.
true, but the nice thing about cauchy sequences is that there's no limits involved. note that:
$\displaystyle \left|\frac{i(m- n)}{mn}\right| = |i|\left|\frac{m-n}{mn}\right| = \left|\frac{m-n}{mn}\right|$
$\displaystyle \leq \left|\frac{1}{n}\right| + \left|\frac{-1}{m}\right| = \left|\frac{1}{n}\right| + \left|\frac{1}{m}\right|$
if we choose $\displaystyle N > 2/\epsilon$, then for m,n > N
$\displaystyle \left|\frac{1}{n}\right| + \left|\frac{1}{m}\right| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$
NOTE: if i represents a constant, rather than the square root of -1, then just choose:
$\displaystyle N > \frac{2|i|}{\epsilon}$
Deveno you misread slightly (it was i^m not mi) but the proof amounted to the same thing. Let e>0 be given. $\displaystyle |\frac{i^n}{n}-\frac{i^m}{m}|<_ |\frac{i^n}{n}|+|\frac{i^m}{m}|=\frac{1}{n}+\frac{ 1}{m}$ Then choosing N >2/e completes the proof.