\textbf{Problem 2, Part 3: Linear recurrence sequences with variable coefficients}

We consider a sequence $\left( v _ { n } \right) _ { n \geqslant 0 }$ satisfying a recurrence of the form
$$v _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } b _ { i } ( n ) v _ { n + i }$$
where $v _ { 0 } , \ldots , v _ { d - 1 }$ are given and for all $i \in \{ 0 , \ldots , d - 1 \} , \left( b _ { i } ( n ) \right) _ { n \geqslant 0 }$ is a sequence with complex values converging to $a _ { i }$. We also define for all $n \geqslant 0 , V _ { n } = \left( v _ { n } , \ldots , v _ { n + d - 1 } \right)$. We always assume hypothesis (*) is satisfied.

Deduce that $v _ { n }$ tends to 0.