Let $\{f_n\}$ be a sequence of continuous functions such that$ f_n \to f$ uniformly on $\mathbb R$. Suppose that $x_n\to x_0$. Prove that $\lim_{n\to\infty} f_n(x_n)=f(x_0)$.
Let me know if I should format it better! This question seems so straightforward that I don't know what definition I'm supposed to employ...
Let $\;\epsilon >0\;$:
$$|f_n(x_n)-f(x_0)|=|f_n(x_n)-f(x_n)+f(x_n)-f(x_0)|\le$$
$$|f_n(x_n)-f(x_n)|+|f(x_n)-f(x_0)|$$
For the first summand in the second line above there exists $\;N_1\in\Bbb N\;$ s.t. it is less than $\;\frac\epsilon2\;$ for any $\;x_n\;$ as long as $\;n>N_1\;$, and for the second summand there exists $\;N_2\in\Bbb N\;$ s.t. it is less than $\;\frac\epsilon2\;$ as long as $\;n>N_2\;$ by continuity (why?) ...and we're done.