A question about the convergence of an infinite product in Complex Analysis

Question

In the book "Theory of Functions of a Complex Variable" by A. I. Markushevich, the extended definition of convergence of an infinite product is as follows: Given an infinite product $ \mathop{\prod}_{n=1}^{\infty} u_n $ with finitely many zero factors, let $ N \ge 0 $ be any integer such that all the factors $ u_{N+1} ,u_{N+2},... $ are nonzero, and let $ v $ be any integer greater than $ N $, then we say that $ \mathop{\prod}_{n=1}^{\infty} u_n $ converges if the limit $$ \lim_{v \to \infty} \mathop{\prod}_{n=1}^v u_n = \lim_{v \to \infty} \mathop{\prod}_{n=1}^N u_n \cdot \mathop{\prod}_{n=N+1}^v u_n = \mathop{\prod}_{n=1}^N u_n \cdot \lim_{v \to \infty} \mathop{\prod}_{n=N+1}^v u_n$$ exists. From this definition, the author assert that "an infinite product vanishes if and only if at least one of its factors vanishes." Is infinite product $ \mathop{\prod}_{n=1}^{\infty} \frac 1n $ a counterexample to this assertion?