finding certain sequences that satisfy a requirement I need to find sequence $(z_n) $ and $(w_n)$ such that $|z_n| \to 1 $ and $|w_n | \to 1 $ but
$$ \Big| \frac{ w_n - z_n }{1- \overline{w}_n z_n } \Big | \; \; \text{doesn't converge to} \; \;  1 $$
My try
Put $z_n = 1 + \frac{1}{n}$ and $w_n = 1 - \frac{1}{n} $, then $|z_n| = | 1 + \frac{1}{n} | \to 1 $ and $|w_n| \to 1 $, but
$$ \Big| \frac{ w_n - z_n }{1- \overline{w}_n z_n } \Big | = \Big| \frac{ - \frac{2}{n}}{1 - (1^2 - \frac{1}{n^2})} \Big| = \Big| \frac{ - \frac{2}{n} }{\frac{1}{n^2}} \Big| = 2n$$
which does not converge to $1$ as required. My question is, is this a correct solution? What are all possible limits of such sequences?

For a much simpler example, take
$$z_n=w_n=1$$
for all $n$. Clearly, both $|z_n|\rightarrow1$ and $|w_n|\rightarrow1$, but
$$\Big| \frac{ w_n - z_n }{1- \overline{w}_n z_n } \Big |=\Big| \frac{0}{0} \Big |$$
is undefined and in particular does not converge to $1$, as desired.
