Solution
= Solution
Boundedness of $(z_n)$ makes both real sequences $(x_n)$ and $(y_n)$ bounded. The <Bolzano-Weierstrass theorem> first gives a subsequence $(z_{n_j})$ for which $(x_{n_j})$ converges. Applying it again to $(y_{n_j})$ gives a subsubsequence for which the imaginary parts also converge. Since convergence of <complex numbers> is equivalent to convergence of their <real parts> and <imaginary parts>, this subsubsequence converges in $\mathbb C$.