The limit of complex sequence $$\lim\limits_{n \rightarrow \infty} \left(\frac{i}{1+i}\right)^n$$
I think the limit is $0$; is it true that $\forall a,b\in \Bbb C$, if $|a|<|b|$ then $\lim\limits_{n\rightarrow \infty}\left(\frac{a}{b}\right)^n=0$?
I would like to see a proof, if possible. Thank you

You are right.
For a proof, just observe that if $\frac{|a|}{|b|}<1$, then
$$
\left\|\left(\frac ab\right)^n\right\|
=
\left(\frac{|a|}{|b|}\right)^n
\to
0
$$
as $n\to\infty$.
