# How to prove the following properties of infimum and supremum involving the union and intersection of the sets $A_k$

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

• Thanks for the quick reply, I was suspecting this. What I don't understand is the $k\ge n$ part. Say I was given $N$ sets $A_k$, $k = 1..N$, is it correct to say that $\sup A_{N-5} = \bigcup_{k=1}^{N-5}A_k$. – Alexander Cska Jan 9 '16 at 10:29
• @AlexanderCska: That is a different notation question. I would have thought that $A_{N-5}$ would be a particular set given $N$ so $\sup A_{N-5}$ is not particularly helpful (it is the same set). More interesting might be $\lim \sup$ and $\lim \inf$ (see Wikipedia for its use of notation) – Henry Jan 9 '16 at 10:53
• If I am not mistaken, $\lim \sup A_k = \bigcap_{k=1} \bigcup_{m=k} A_m=\inf_{n\ge 1} (\sup_{m\ge n} A_m)$, which I figured out by reverse engineering the definition form my question. This is actually $\inf_{k\ge 1}(\sup_{m\ge k} A_m)$. In my opinion due to the intersection, we generate a monotonically decreasing sequence $\lim_{k\to \infty} \downarrow \sup_{m\ge k} A_m$ – Alexander Cska Jan 9 '16 at 11:08