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Put $U_0=1$, so the same definition $z_n=x_n/U_{n-1}$ includes $n=1$. Since $U_n=a_n{U}_{n-1}$ and every $a_n$ is nonzero,The factor $U_n^{-1}$ is a discrete integrating factor for this variable-coefficient affine recurrence. Sum the increments as a telescoping series and multiply back by $U_n$:The identity requires no assumption that the coefficients have the same sign.

A variable-coefficient affine recurrence is a linear recurrence relation with coefficients allowed to depend on the index. If every $a_n$ is nonzero, a discrete integrating factor reduces it to a telescoping series.