Is $A$ a convex set of $\mathbb{C}$?

Question

Let us consider the following set \begin{eqnarray*} A &=&\Big\{\lambda\in \mathbb{C};\;\exists x_n=(a_n,b_n)\in \mathbb{C}^2\,;\;\;\;|a_n+b_n|=1, \\ &&\phantom{++++++++++}\;\displaystyle\lim_{n\longrightarrow+\infty}|a_n|^2+a_n\overline{b_n}= \lambda\;\;\hbox{and}\;\;\displaystyle\lim_{n\longrightarrow+\infty}|a_n|<\infty\Big\}. \end{eqnarray*} Is $A$ a convex set of $\mathbb{C}$?

Thank you.

Answer 1

Maybe I'm missing something but if I take $a=\lambda$ and $b = 1-\lambda$, then $|a|^2+a\bar{b} = \lambda$, so it looks like this set is the whole plane.