the Blaschke condition
/ BLAHSH-kuh /
If you want to plant infinitely many zeros a_1, a_2, ... inside the unit disk and still have a BOUNDED holomorphic function vanishing exactly there, you cannot crowd them too densely near the boundary circle. The Blaschke condition is the precise quota: the zeros must satisfy sum over n of (1 - |a_n|) < infinity. The quantity 1 - |a_n| is the distance from a_n to the boundary, so the condition says the zeros' distances-to-the-edge must add up to a finite total — they may approach the boundary, but not too eagerly.
This is exactly the convergence criterion for the infinite Blaschke product over those zeros. Each Blaschke factor B_(a_n)(z) deviates from 1 by an amount controlled by 1 - |a_n|, so by the absolute-convergence test for products, the product converges to a nonzero bounded function precisely when sum (1 - |a_n|) is finite. When the condition holds, the Blaschke product gives you the desired bounded function; when it fails, the product converges to 0 identically, and in fact there is NO bounded holomorphic function (not identically zero) with those zeros at all.
The Blaschke condition is the disk's sharp boundary on the design freedom that Weierstrass enjoyed on the whole plane. On the plane any discrete sequence going to infinity is an allowable zero set, with no rate limit. On the disk, boundedness exacts a price: the zeros must thin out fast enough as they approach the rim. The condition is the gateway to the theory of Hardy spaces and the Nevanlinna class — the zero sets of bounded (and more generally, of suitably-restricted) analytic functions on the disk are exactly the Blaschke sequences, those satisfying sum (1 - |a_n|) < infinity.
Zeros at a_n = 1 - 1/n^2 (real, approaching 1) satisfy sum (1 - |a_n|) = sum 1/n^2 < infinity, so they form a Blaschke sequence and a bounded function vanishing there exists. But zeros at a_n = 1 - 1/n give sum (1 - |a_n|) = sum 1/n = infinity, violating the condition — they approach the boundary too quickly, and no bounded nonzero holomorphic function can have exactly these zeros.
1 - 1/n^2 passes the Blaschke test (sum 1/n^2 finite); 1 - 1/n fails it (sum 1/n diverges).
The condition constrains only how the zeros approach the boundary; any finite set, or zeros that stay in a smaller disk |z| <= r < 1, trivially satisfy it. The whole subtlety is the rate at which |a_n| -> 1, measured by summing the gaps 1 - |a_n|.