# Question about applying a sequence of continuous functions that converge uniformly to a convergent sequence of complex numbers

Let $$(w_n)$$ be a sequence of complex numbers in the closed unit disk $$D = \{ z \in \mathbb{C} \ \ | \ \ |z| \leq 1 \}$$ that converges to $$w$$. Suppose that $$(f_n)$$ is a sequence of continuous functions from $$D$$ to $$\mathbb{C}$$ that converges uniformly on $$D$$ to a function $$f$$. Prove or disprove that $$\lim_n f_n(w_n) = f(w).$$

My working so far: On the expectation that the statement is true, I have tried attacking it directly using the usual $$\epsilon$$'s and $$N$$'s, but to no avail. I have also tried to show that $$(f_n(z_n))$$ is a Cauchy sequence but that does not seem to work either.

Let us fix $$\epsilon > 0$$.

We want to prove, that there exists $$N$$, such that for all $$n>N$$ we have $$|f_n(\omega_n) - f(\omega)| < \epsilon$$

You know, that since $$f_n$$ converges to $$f$$ on D, and $$\omega \in D$$, there exists $$N_f$$, such that for all $$n>N_f$$, we have $$|f_n(\omega) - f(\omega)| < \frac{\epsilon}{2}$$ $$\ \ \ \ \ (*)$$

Similarly, we want to show $$|f_n(\omega_n) - f_n(\omega)| < \frac{\epsilon}{2}$$ for $$n>N_\omega$$. Why does there exist such a $$N_\omega$$ ?. Because, we know that $$\forall_n$$ $$f_n$$ is a continuous function on D and $$\omega_n$$ converges to $$\omega$$ (so, there exists $$N_\omega$$ that for every $$n>N_\omega$$ we have $$|\omega_n - \omega| < \delta_\omega \ \$$ (for particular $$\delta_\omega$$), and now by continuity of $$f_n$$, since $$|\omega_n - \omega| < \delta_\omega$$, we get $$|f_n(\omega_n) - f_n(\omega)| < \frac{\epsilon}{2}$$ $$\ \ \ \ \ \ \ (**)$$

Now, taking $$N = max\{N_f,N_\omega\}$$, we get that $$\forall_{n>N}$$ $$|f_n(\omega_n) - f(\omega)| = |f_n(\omega_n) - f_n(\omega) + f_n(\omega) - f(\omega)| <^{triangle}_{inequality} \\ < |f_n(\omega_n) - f_n(\omega)| + |f_n(\omega) - f(\omega)| <^{(*)}_{(**)} \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$$

Q.E.D

Hint: Using the $$\varepsilon$$-$$\delta$$ definition is a good way to go. Here's an inequality that'll help you:$$|f_n(w_n)-f(w)|\le|f_n(w_n)-f(w_n)|+|f(w_n)-f(w)|.$$