Suppose $f_{n} \to f$ uniformly on a set $E$ in a metric space. Let $x$ be a limit point of $E$, and suppose that $$ \lim_{ t \to x } f_{n}(t) = A_{n} \quad (n = 1, 2, 3, \dots). $$ Then $\{ A_{n} \}$ converges, and $$ \lim_{ t \to x } f(t) = \lim_{ n \to \infty } A_{n}. $$ In other words, the conclusion is that $$ \lim_{ t \to x } \lim_{ n \to \infty } f_{n}(t) = \lim_{ n \to \infty } \lim_{ t \to x } f_{n}(t). $$