I am reading a book on probability theory and I have troubles understanding why the following holds
$$ \sup_{k \ge n} A_k = \bigcup_{k\ge n} A_k $$
$$ \inf_{k \ge n} A_k = \bigcap_{k\ge n} A_k $$
I know how to define infimum and supremum, but I have troubles proving the above expressions. Unfortunately, the book states those expressions as a definition without proving them. A graphical explanation in terms of a Venn diagram will be very helpful. Thank you in advance