In the last section, we learned about contour integrals. A fundamental theorem of complex analysis concerns contour integrals, and this is Cauchy's theorem, namely that if is holomorphic, and the domain of definition of has somehow the right shape, then
for any contour which is closed, that is, (the closed contours look a bit like a loop). For this theorem to hold, surprisingly, the shape of the domain of definition is supremely important; for some it does hold, for some it doesn't. In this chapter, we will prove that the theorem holds for certain which are so-called star-shaped domains. Later on in the book, we will see that it even holds for a larger class of domains, namely the simply connected ones, which will require advanced tools which we will build up along the course of this book.
Let be a sequence of functions defined on an interval which is
equicontinuous (that is, for any there exists such that ) and
uniformly bounded (that is, there exists such that ).
Then contains a uniformly convergent subsequence.
Let be an enumeration of the set . The set is bounded, and hence has a convergent subsequence due to the Heine–Borel theorem. Now the sequence also has a convergent subsequence , and successively we may define in that way.
Set for all . We claim that the sequence is uniformly convergent. Indeed, let be arbitrary and let such that .
Let be sufficiently large that if we order ascendingly, the maximum difference between successive elements is less than (possible since is dense in ).
Let be sufficiently large that for all and .
Set , and let . Let be arbitrary. Choose such that (possible due to the choice of ). Due to the choice of , the choice of and the triangle inequality we get
Hence, we have a Cauchy sequence, which converges due to the completeness of .