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
These are the definitions of supremum and infimum as applied to sets rather than numbers.
With numbers, a supremum is the least upper bound, i.e. the smallest number greater than or equal to the specified numbers. With sets, it is the smallest set which has all the specified sets as subsets.
Similarly with numbers, an infimum is the greatest lower bound, i.e. the largest number less than or equal to the specified numbers. With sets, it is the largest set which is a subset of all the specified sets.