Let us assume now that $\{X_t\}$ is a stationary time series with $$X_t=\phi X_{t-1}+Z_t \qquad t=0,\pm 1,\dots,$$ where $\{Z_t\}\sim > WN(0,\sigma^2)$ (White noise), $|\phi|<1$ and $\{Z_t\}$ is uncorrelated with $X_s$ for each $s<t$. Show that $EX_t=0$.
What I did is $$E[X_t]=E[\phi X_{t-1}+Z_t]\Rightarrow E[X_t]=E[\phi X_{t-1}]$$ and
$$E[X_t]=\phi E[X_{t-1}]\Leftrightarrow E[X_t]=E[X_{t-1}]=0$$
because if a time series $\{X_t\}$ is stationary then $\mu_X(t)$ is independent of $t$.
Is that right?