autocorrelation - Does the auto-correlation function of stationary random process always converge?

The auto-correlation function of the stationary random process only depends on the time difference $\tau$.

http://web.ntpu.edu.tw/~yshan/chapter6_han.pdf

64th slide of this lecture note mentions that for a zero-mean process, the autocorrelation converges to zero as $\tau$ goes to infinity.

I want to know why.