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

Observe that $$\frac{i}{1+i} =\frac{1}{2}+\frac{i}{2}$$ Now define $$F_n=\Big(\frac{i}{1+i}\Big)^n$$ Then $$F_1=\frac{1}{2}+\frac{i}{2}$$ $$F_2=\frac{i}{2}$$ $$F_3=-\frac{1}{4}+\frac{i}{4}$$ $$F_4=-\frac{1}{4}$$ $$F_5=-\frac{1}{8}-\frac{i}{8}$$
