Sequence of sequences/Product sequence with growing number of factors/Exercise

Suppose that for every natural number ${}k$, a null sequence ${}y_{k}$ is given, we denote the ${}n$-th member of the ${}k$-th sequence by ${}y_{kn}$. Is the sequence ${}z_{n}$, whose ${}n$-th member is defined by

${}z_{n}=\prod _{k=1}^{n}y_{kn}\,,$

also a null sequence?

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