the uniform convergence of a product
When the factors of a product depend on z, we do not just want the product to converge for each separate z — we want it to converge in a way that respects holomorphy. The right strength is uniform convergence on compact sets: on each closed bounded piece of the region, the partial products approach the limit at a single rate that does not depend on which z you picked. This is exactly the condition that lets the limit inherit holomorphy from its factors (uniform limits of holomorphic functions are holomorphic, by Weierstrass).
Here is the practical handle. Suppose the f_n are holomorphic on a region and on every compact subset K you can bound sum (sup over K of |f_n|) < infinity — that is, the deviations are uniformly summable on K. Then product of (1 + f_n) converges uniformly on K, the limit is holomorphic, and you can differentiate or integrate the product by working with the factors. This 'uniform summability of the deviations on compacta' is sometimes called normal convergence, and it is the standard hypothesis under which every Weierstrass and Mittag-Leffler construction is carried out.
Why insist on compact sets rather than the whole region at once? Because functions like the sine product have factors whose deviations do NOT stay uniformly small near infinity — sum |z^2/n^2| blows up as |z| grows. The cure is to fix attention on each bounded disk |z| <= R at a time: on that disk the tail of the sum is uniformly tiny, so the product behaves perfectly. Letting R grow then covers the whole plane disk by disk. This 'locally uniform' viewpoint is the standard and honest way to handle products of functions on unbounded domains.
For the product of (1 - z^2/n^2), fix a disk |z| <= R. There sup of |z^2/n^2| over the disk is R^2/n^2, and sum R^2/n^2 = R^2 pi^2/6 < infinity. So the deviations are uniformly summable on the disk, the product converges uniformly there, and the limit (a multiple of sin) is holomorphic on |z| <= R. Since R was arbitrary, sin is entire.
Bound the deviations on each disk |z| <= R by a convergent number series; uniform convergence on every disk gives an entire limit.
Pointwise convergence of a product of holomorphic functions is not enough to guarantee a holomorphic limit; you need uniformity on compacta. This is the same lesson as for series of functions — convergence at each point alone does not transmit smoothness.