I just can't understand the concept of $\lim \sup/\inf$ of sets, even after reading a lot about it. In limit infinum why do we have union sign with intersections (infima)? Can someone explain it very easily? Thank you very much.
Suppose you have a sequence that represents a function of time. Suppose further that this sequence is the sum of two separate contributions: transient and steady state. The transient part of the signal is the part that has a limited time presence. The steady state part of the signal is the part that goes on forever.
For example, consider the following function $ f(x) = exp( -x / 10 ) + sin( x )$
Note that the exponential part of the signal dies away to $0$, while the sin part of the signal stays around forever.
Intuitively, limit superior and limit inferior are asking what are the supremum and infimum after all of the transient parts have died away.