The right notion of convergence
This rung is going to build toward the crown jewel of the whole subject, the Riemann mapping theorem, and the engine that powers it is a machine for taking limits of holomorphic functions. So the very first question we must settle is which kind of convergence we should even ask for. Pointwise convergence — f_n(z) approaches f(z) at each individual point z — is far too weak; it lets all sorts of pathologies through and tells you nothing about derivatives. The strongest natural choice, uniform convergence everywhere on the region at once, is too strong: most useful sequences fail it near the boundary, where the functions can grow without bound.
The Goldilocks notion is locally uniform convergence, also called uniform convergence on compact subsets. It says: for every compact set K sitting inside the region, the convergence f_n approaches f is uniform on K. Equivalently, every point of the region has a small closed disk around it on which the convergence is uniform. You met this exact idea one rung earlier, where you proved that a power series converges uniformly on every compact subset of its disk of convergence — even though it can fail to converge uniformly on the whole open disk. Locally uniform is the natural home for everything holomorphic.
The theorem, and why it should astonish you
Here is the headline, Weierstrass's convergence theorem. Suppose each f_n is holomorphic on an open region D, and suppose f_n approaches a limit f locally uniformly on D. Then f is holomorphic on D — full stop. And there is a second half that is almost more remarkable than the first: the derivatives converge too. The sequence f_n' approaches f' locally uniformly, and the same holds for every higher derivative. Differentiation, an operation that usually destroys convergence, sails right through the limit.
To feel how strange this is, you have to remember how badly the real line behaves. On the reals, a uniform limit of smooth functions need not be differentiable at all. The classic shock is that every continuous function on an interval — including the nowhere-differentiable Weierstrass monster, a curve that wiggles at every scale — is a uniform limit of polynomials. Polynomials are as smooth as anything could be, yet their uniform limit can have no derivative anywhere. Real uniform convergence simply does not protect smoothness, and it certainly does not let you swap limit and derivative.
So Weierstrass's theorem is not a complex echo of some real-variable fact — there is no real-variable parallel. It is yet another face of the rigidity you have been meeting all along: complex differentiability is so demanding a condition that it is preserved by limits, while real differentiability is so flimsy it is destroyed by them. The whole reason this works is that being holomorphic is secretly an integral condition, not a pointwise differential one, and integrals are exactly what uniform convergence is good at passing through.
The proof: Morera does it in one line
The cleanest proof leans on Morera's theorem, the converse to Cauchy's theorem you proved last rung: if a continuous function g has the property that its integral around every small triangle (or closed loop) in the region is zero, then g is automatically holomorphic. Morera turns holomorphy from a statement about derivatives into a statement about vanishing integrals — and integrals are precisely where uniform convergence is harmless. That single swap of perspective is what makes the whole argument trivial.
- First, the limit f is continuous. A locally uniform limit of continuous functions is continuous, and each f_n is continuous (being holomorphic), so f is continuous on D. We have earned the right to integrate it.
- Pick any closed triangle T whose inside lies entirely in D. Because each f_n is holomorphic, Cauchy's theorem gives that the integral of f_n around the boundary of T is zero, for every single n.
- Now let n go to infinity. The triangle's boundary is a compact set, so f_n approaches f uniformly there, and uniform convergence lets you swap the limit with the integral: the integral of f around the boundary of T equals the limit of the integrals of f_n, which is the limit of a sequence of zeros — namely zero.
- So f is continuous and integrates to zero around every triangle in D. By Morera's theorem, f is holomorphic. That is the whole proof of the first half.
Notice what carried the whole load: the swap in step three, where uniform convergence on the compact loop let the limit pass inside the integral, turning a limit of zeros into the integral of the limit being zero. That move is illegal for pointwise convergence and dangerous for derivatives, but it is the everyday bread and butter of uniform convergence over integrals. Morera supplied the bridge from integrals back to holomorphy; uniform convergence did the crossing.
The bonus: derivatives converge for free
The second half — that f_n' approaches f' — comes from the same source of magic that powered the last few rungs: the Cauchy integral formula for derivatives. Recall that it expresses a derivative not as a limit of difference quotients but as an integral over a surrounding circle. Writing it out for the gap between the n-th derivative and the limit's derivative makes the convergence almost self-evident.
f_n'(z_0) - f'(z_0)
= (1 / 2 pi i) integral over |w - z_0| = r of
[ f_n(w) - f(w) ] / (w - z_0)^2 dw
|f_n'(z_0) - f'(z_0)| <= (1/r) * max over the circle of |f_n - f|
the max -> 0 (uniform convergence on the circle) => f_n'(z_0) -> f'(z_0)Read the estimate slowly, because it is doing all the work. Differentiation in the usual difference-quotient picture is dangerous: dividing by a tiny step can wildly amplify a small error in the function. But the Cauchy formula divides instead by (w - z_0)^2 on a circle of a fixed, comfortable radius r — never by anything small. So a small uniform error in f_n becomes a small error in f_n', controlled by the same supremum scaled only by 1/r. This is the deep reason swapping limit and derivative is legal here: in the complex world, the derivative is an averaging integral, and averaging tames errors instead of amplifying them. These bounds are the very same Cauchy estimates you used to prove Liouville's theorem.
Why this is the foundation of the whole rung
It is hard to overstate how much this little theorem unlocks. The plan for reaching the Riemann mapping theorem is a compactness argument: we will build a sequence of holomorphic candidate maps, extract a convergent subsequence, and crown its limit as the answer. That strategy is dead on arrival unless the limit of holomorphic functions is itself holomorphic — and Weierstrass's theorem is exactly the promise that the limit stays inside the club. A normal family, the central object of the next guide, is by definition a family of holomorphic functions from which every sequence has a locally uniformly convergent subsequence; this theorem is what guarantees those limits are usable.
The derivative half pays off just as directly. To prove the Riemann map is injective we will need the limit of injective maps to stay injective, and that is the job of Hurwitz's theorem in a couple of guides' time — a result that argues by counting zeros and so leans on the fact that f_n' and f' travel together. Even Vitali's theorem, which upgrades convergence on a small set to convergence everywhere, is a refinement that takes Weierstrass as its starting point. The honest summary: almost every limiting argument later in this rung quietly cites the theorem you just proved.