an infinite product of holomorphic functions
Now let the factors be functions, not numbers. We want to multiply infinitely many holomorphic functions together — write them as 1 + f_n(z) — and ask whether the product P(z) = product of (1 + f_n(z)) is itself a sensible holomorphic function of z. This is the engine room of construction: it is how we will manufacture functions with a prescribed set of zeros, by choosing factors that each vanish where we want a zero.
The clean theorem: if the f_n are holomorphic on a region and the series sum |f_n(z)| converges UNIFORMLY on every compact subset of the region, then the product P(z) converges uniformly on compacta and the limit P is holomorphic there. Even better, the zeros of P are exactly the points that are zeros of at least one factor 1 + f_n, and the multiplicity (order) of a zero of P at a point is the sum of the orders contributed by the individual factors that vanish there. So the product literally assembles a function out of its zeros: each factor donates a zero, and the limit collects them all.
There is a beautiful bonus formula for the logarithmic derivative. Differentiating log P = sum log(1 + f_n) term by term gives P'(z)/P(z) = sum f_n'(z) / (1 + f_n(z)), valid where the sum converges nicely. This turns a product into a sum of simple pieces and is the bridge to partial-fraction expansions: differentiating the sine product, for example, hands you the cotangent's partial-fraction expansion directly. Products and partial fractions are two views of the same object, joined by the logarithmic derivative.
The product P(z) = product over n >= 1 of (1 - z^2 / n^2) has factors 1 + f_n with f_n(z) = -z^2/n^2, and sum |f_n(z)| = |z|^2 sum 1/n^2 converges uniformly on any disk |z| <= R. So P is entire, and it vanishes exactly where some factor does — at z = +-n, the nonzero integers. Up to the factor pi z, this is sin(pi z).
Uniform convergence of sum |f_n| on compacta makes the product holomorphic; its zeros are collected from the factors — here the integers.
It is sum |f_n(z)|, not the product of |1 + f_n|, that must converge uniformly. Demanding the moduli of the factors converge is the wrong test; the right object is the deviations f_n, just as for products of numbers it was the a_n.