Q:

Holomorphic bounded function with zeros in $1-1/n$

The question is:

Let $f$ be holomorphic function in unit disc, which is bounded and
  have zeros in points $1-1/n$ for natural $n$. Then $f$ is equal to
  zero.

Identity theorem does not help because point 1 is not inside unit disc. I suppose it follows from the fact that any holomorphic function in unit disc is either unbounded or may be extended on larger disc, but I can not prove it. Is it correct?

A:

This follows from Blaschke's condition: the sequence $(a_n)$ is the zero sequence of a bounded holomorphic function on the unit disc if and only if
$$
\sum_n (1-|a_n|) < \infty.
$$
The simplest proof is via Jensen's formula.

As noted, this doesn't follow from the identity theorem: there are unbounded holomorphic functions on the unit disc whose zero sequence is $(1-1/n)$.

