Editing Morera's theorem (section)

===Uniform limits===
For example, suppose that ''f''<sub>1</sub>,&nbsp;''f''<sub>2</sub>,&nbsp;... is a sequence of holomorphic functions, [[uniform convergence|converging uniformly]] to a continuous function ''f'' on an open disc.  By [[Cauchy's integral theorem|Cauchy's theorem]], we know that
<math display="block">\oint_C f_n(z)\,dz = 0</math>
for every ''n'', along any closed curve ''C'' in the disc.  Then the uniform convergence implies that
<math display="block">\oint_C f(z)\,dz = \oint_C \lim_{n\to \infty} f_n(z)\,dz =\lim_{n\to \infty} \oint_C f_n(z)\,dz = 0 </math>
for every closed curve ''C'', and therefore by Morera's theorem ''f'' must be holomorphic.  This fact can be used to show that, for any [[open set]] {{math|Ω ⊆ '''C'''}}, the set {{math|''A''(Ω)}} of all [[bounded function|bounded]], analytic functions {{math|''u'' : Ω → '''C'''}} is a [[Banach space]] with respect to the [[supremum norm]].