Consider the definition of $m$-dependence:
A sequence of random variables ${(X_t)}_{t \in \mathbb{Z}}$ is said to be $m$-dependent for $m \in \mathbb{N}_{0}$ if for each $t$ ${(X_s)}_{s \leq t}$ is independent of ${(X_s)}_{s \geq t+m+1}$.
Now let ${(Z_t)}_{t \in \mathbb{Z}}$ be an i.i.d. sequence of random variables, and define $$X_t := Z_t+\theta_1 Z_{t-1} + \ldots + \theta_m Z_{t-m}$$ with $\theta_1, \ldots, \theta_q \in \mathbb{R}$.
How can one show that ${(X_t)}_{t \in \mathbb{Z}}$ is $m$-dependent?
My reasoning:
Consider $X_r$ and $X_s$ with $r -s\geq m+1$. Then
$$ X_r = Z_r+\theta_1 Z_{r-1} + \ldots + \theta_r Z_{r-m}, $$ $$ X_s = Z_s+\theta_1 Z_{s-1} + \ldots + \theta_s Z_{s-m}. $$ Since $\sigma(Z_s, \ldots, Z_{s-m}) \perp \sigma(Z_r, \ldots, Z_{r-m})$, then $X_r$ and $X_s$ are also independent whenever $r -s\geq m+1$.
But how can one show that for every $t \in Z$ it holds $$ \sigma({(X_s)}_{s \leq t}) \perp \sigma({(X_s)}_{s \geq m + t +1})\ ?$$