the Weierstrass convergence theorem
/ VY-er-shtrahss /
On the real line, a uniform limit of differentiable functions can be a disaster: the limit need not be differentiable at all, and even when it is, the derivatives can fail to converge to its derivative. Weierstrass's classic 'continuous-but-nowhere-differentiable' function is built as exactly such a limit. The complex world is dramatically better behaved, and this theorem is the reason. It says that if holomorphic functions converge in the locally uniform sense, the limit is automatically holomorphic, and the whole tower of derivatives converges along with it.
Precisely: let f_n be holomorphic on an open region D and suppose f_n -> f locally uniformly (uniformly on every compact subset). Then f is holomorphic on D, and for each order k the derivatives f_n^(k) converge to f^(k), again locally uniformly. The proof is a beautiful one-line use of Cauchy's machinery. First, f is continuous as a locally uniform limit of continuous functions. Then for any small loop the integral of f around it is the limit of the integrals of f_n, each of which is 0 by Cauchy's theorem, so f integrates to 0 around every loop and is holomorphic by Morera's theorem. For the derivatives, write f_n^(k)(z) using the Cauchy integral formula as a contour integral of f_n divided by (w - z)^(k+1); the uniform convergence on the fixed contour lets you pass the limit through the integral, giving f^(k) and its convergence.
This single result is the engine of the whole subject of normal families. It guarantees that the limits produced by Montel's compactness arguments — and in particular the extremal map in the Riemann mapping theorem — are themselves holomorphic, so the limiting object lives in the right space. It also justifies differentiating power series and infinite products term by term inside their region of convergence. The honest caveat: you really do need locally uniform convergence; mere pointwise convergence of holomorphic functions can produce a non-holomorphic (even discontinuous) limit.
Each partial sum of the exponential series, P_N(z) = 1 + z + z^2/2! + ... + z^N/N!, is a polynomial, hence entire. The sums converge locally uniformly to e^z. By Weierstrass, the limit e^z is entire and its derivatives match: differentiating term by term gives the same series, recovering (e^z)' = e^z.
A locally uniform limit of entire functions is entire, and the derivatives converge to the derivative.
The striking part is the convergence of all derivatives, which has no counterpart for real differentiable functions — there a uniform limit need not even be once differentiable. Locally uniform convergence cannot be weakened to pointwise.