EEG signal processing with sampling frequency
Intuitively... Assume to have a sampled signal, i.e. a sequence $\{s(n)\}$. The "fastest" oscillatory behaviour you can see is a component which changes "polarity" at any successive sample, for example consider
$$
s(n) = (-1)^n
$$

it is positive for $n=0$, negative for $n=1$, positive again for $n=2$ and so on. Trivially you cannot have something oscillating faster. The period of this signal is 2 (samples), infact you have $s(n) = s(n+2)$ for any $n$. Therefore its frequency is half the sampling frequency.