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


1 Answer 1


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.

  • $\begingroup$ 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 $\sup A_{N-5} = \bigcup_{k=1}^{N-5}A_k$. $\endgroup$ Jan 9, 2016 at 10:29
  • $\begingroup$ @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) $\endgroup$
    – Henry
    Jan 9, 2016 at 10:53
  • $\begingroup$ 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 $\lim_{k\to \infty} \downarrow \sup_{m\ge k} A_m$ $\endgroup$ Jan 9, 2016 at 11:08
  • $\begingroup$ You have a small typo in the second sentence (sumpremum, which sounds like a new definition for some sums). $\endgroup$
    – Olorun
    Jan 11, 2016 at 7:14
  • 1
    $\begingroup$ @Olorun - you are right so I will edit it. My spell checker offerings include supremo or super-mum $\endgroup$
    – Henry
    Jan 11, 2016 at 11:28

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