Blaschke product is defined in the following way $:$
$$B(z) = \prod\limits_{n = 1}^{\infty} \frac {|z_n|} {z_n} \frac {z_n - z} {1 - \overline z_n z},\ z_n \in \mathbb D \setminus \{\textbf 0\}\ \forall n$$
It can be shown that if $\sum\limits_{n = 1}^{\infty} (1 - |z_n|) \lt \infty$ then the sequence of partial products of $B(z)$ converges uniformly for any $z \in \mathbb C$ with $|z| \leq r \lt 1.$ Hence by Morera's theorem $B(z)$ gives rise to a holomorphic function on the open unit disk $\mathbb D.$ Now since $z_n$'s are confined in a bounded set $\mathbb D$ and are zeros of a holomorphic function given by the Blaschke product $B(z)$ it turns out that each $z_n$ has finite multiplicity and hence the range of the sequence $\{z_n \}_{n \geq 1}$ is a bounded infinite subset of $\mathbb C$ and hence by Bolzano-Weierstrass theorem it has an accumulation point. But then the identity theorem asserts that $B(z)$ is identically zero on $\mathbb D.$ Where did I make mistake?
Any suggestion would be greatly appreciated. Thanks for investing your valuable time on my question.
The $z_n$ accumulate at a point on the boundary of the open set $\mathbb D$, not within $\mathbb D$, so the identity theorem does not apply. That theorem requires your function to be analytic in an open neighborhood around your accumulation point.