Problem 1789 (difficulty: 6/10)

Let \(\displaystyle a_1,a_2,\dots\) be a sequence of complex numbers such that \(\displaystyle |a_k|<1\) and \(\displaystyle \re a_k>\frac12\) for all \(\displaystyle k\). Let

\(\displaystyle z_0=0, \qquad z_{n+1}=\frac{z_n+a_n}{1+\overline{a_n}z_n}. \)

Prove that \(\displaystyle a_n\to1\).

(IMC 2011/6 alapján)

Give me another random problem!