If I have a sequence of continuous functions $\{ f_n(x) \}$, all of which are non-negative, satisfying
$$\lim_{n \to \infty} \sup_{x \in A} f_n(x) = \sup_{x \in A} \lim_{n \to \infty} f_n(x) = K < \infty$$
and $f_n(x) \to f(x)$, where $f$ is continuous. Under what additional conditions, if any, may I say $f_n(x)$ converges uniformly to $f(x)$?