Convergent sequence of complex numbers

Question

Let $\{x_n\}$ and $\{y_n\}$ be sequence of complex numbers. If $ \lim\limits_{n \rightarrow \infty} x_n = \lim\limits_{n \rightarrow \infty} y_n $, then is $\lim\limits_{n \rightarrow \infty} \lvert x_n \rvert = \lim\limits_{n \rightarrow \infty} \lvert y_n \rvert $? Is the converse true?

Answer 1

For the first part use: $|\,|x_n|-|y_n|\,| \leq |x_n-y_n|$ and to see that the second statement is false take $x_n=1,y_n=-1$ for all $n$.