I've read that a time series should be a weakly stationary for the Autocorrelation function (ACF) to make sense. The definition for weakly stationary series that I have is that all the observations should have the same mean and variance and that $\mathrm{Cov}(x_t,x_{t-1})$ should be same for all $t$. But ACF for a natural number $h$ is defined to be the correlation of $x_t$ and $x_{t-h}$. So, for it to be defined as a function, we should have $\mathrm{Cov}(x_t,x_{t-h})$ independent of $t$. How is this implied by $\mathrm{Cov}(x_t,x_{t-1})$ is independent of $t$??
$\mathrm{Cov}(x_t,x_{t-1})\text{ is independent of }t\implies\mathrm{Cov}(x_t,x_{t-h}) \text{ is independent of } t\,\forall h\in\Bbb N$??
Please help.