Does the auto-correlation function of stationary random process always converge?
No it does not necessarily.

For example the following discrete-time, WSS random process 

$$x[n] = A \sin(\omega_0 n + \phi) $$

which is called the random phase sinusoid, where $A$ and $\omega_0 \neq 0$ are fixed values and $\phi$ is a random variable uniformly distributed in $\phi \in [-\pi,\pi)$ has an auto-correlation function of the form

$$ r_{xx}[k] = \frac{A^2}{2} \cos(\omega_0 k) $$

which does not go to zero as $k$ goes to infinity; $ \lim_{k \to \infty} r_{xx}[k] \neq 0$.

Similarly for a continuous-time process, the same can be shown.

Note, however, that as MattL indicated in his answer as well, the information contained within a random process is mostly included in its innovations part, this is also expressed in Wold decomposition theorem that any random process can be broken into two parts as a predictable periodic part and a regular unpredictable part (which is the innovations part), then if a WSS random process only includes a regular part but no predictable part, then its covariance (or correlation if zero mean) sequence shall go to zero as the lag goes to infinity if it has a uniformly convergent power spectral density; i.e., the DTFT of auto-correlation sequence (ACS) should converge, and this requires the sum of abs value of ACS be finite and this also requires that $\lim_{k \to \infty} r_{xx}[k] = 0$ for a zero-mean, WSS process containing no predictable (periodic) parts in it. 

Furthermore, when the concept of ergodicity is introduced, then one of the necessary conditions for a WSS random process to be ergodic in the mean is that its auto-covariance (or equivalently auto-correlation for a zero mean process) should go to zero as $k$ goes to infinity. May be that was stated in that document.