Time series - showing the sum of an autocovariance is less than infinity

Let $Z_t = \mu + \psi (B) a_t$ where $\psi (B) = \Sigma^{\infty}_0 \psi_j B^j$ , $\psi_0 = 1$ , $\Sigma^{\infty}_0 |\psi_j|<\infty$ and the $a_t$ are white noise with mean 0 and variance $\sigma^2_a$ .Let $\gamma_j$ be the jth autocovariance of the process Zt.

Prove that $\Sigma^{\infty}_{-\infty}|\gamma_j|< \infty$

I know that $|\gamma_j| = E[(Z_t - \mu)(Z_{t+k} - \mu)] <= \sqrt{Var(Z_t)Var(Z_{t+k})} = \sigma^2_a \Sigma \psi^2_j < \infty$

Don't know where to go from there.