Uniform convergence theorem

In the field of mathematics, there are many theorems labeled "uniform convergence." Here we list a few of them.

• If f_n is a sequence of continuous functions converging uniformly to a function f, then f is also continuous.
• If f_n is a sequence of Riemann integrable functions of an interval [a,b] converging uniformly to a function f, then f is also Riemann integrable and ∫f_n converges to ∫f.
• If f_n is a sequence of holomorphic functions converging uniformly to a function f, then f is holomorphic.
• (Uniform convergence on compacta) If f_n is a sequence of holomorphic functions converging to f, and if the convergence is uniform over any compact set, then f is holomorphic.