Let $(K_n)_{n \in \mathbb N}$ be a sequence of subsets of a metric space $(X,d)$. We say the subset
$$ \limsup_{n \to\infty} K_n := \left\{ x \in X \;|\; \liminf_{n\to\infty} d(x,K_n) = 0 \right\}, $$
$$ \liminf_{n \to\infty} K_n := \left\{ x \in X \;|\; \lim_{n\to\infty} d(x,K_n) = 0 \right\}, $$
where $d(x,K_n) := \inf_{y \in K_n} d(x,y)$.
Note: The above definitions are from the book "Set-valued Analysis" by Jean-Pierre Aubin and Helene Frankowska.
I can't wrap my head around these when I compare them with the alternative definitions of
$$ \limsup_{n \to\infty} K_n := \bigcap_{N=1}^{\infty} \bigcup_{n \geq N} K_n $$
and
$$ \liminf_{n \to\infty} K_n := \bigcup_{N=1}^{\infty} \bigcap_{n \geq N} K_n, $$ which are much easier to intepret.
Can someone provide some explanation or clarifications on how to interpret the first set of definitions and how to relate the two sets of definitions?