# Question about notations involving $\sup$ and $\inf$

This question may be far too basic, but I just want to make sure I understand some notation.

What is the meaning of $\sup_{n} f_n(x)$? Does this mean the largest value taken on by $f_n$ or does it mean the largest value taken on over ALL of the $f_n$ up to $n$ ?

The book then goes on to say $\{ \sup_n f_n > a \} = \bigcup_n \{f_n > a \}$

I just need some help understanding this notation in conceptually.

## 2 Answers

$$\sup_n f_n(x) = \sup\{ f_n(x) : n\in\text{some set} \}$$ where "some set" is usually either $\{1,2,3,\ldots\}$ or $\{0,1,2,3,\ldots\}$. Thus it is $$\sup\{f_1(x),f_2(x),f_3(x),\ldots\}.$$ Its value depends on $x$. If there is a largest one of the members of this set, then it is that one; otherwise it is the smallest number that none of them exceed.

• Thank you. I am still a bit confused about $\{\sup_n f_n > a \} = \bigcup_n \{ f_n > a\}$. Does $\{\sup_n f_n > a \}$ mean that we consider the supremum only of those $f_n$ where $f_n(x) > a$? – Yuugi Sep 20 '14 at 22:35
• It may be that someone used the notation $\{\sup_n f_n>a\}$ to mean the set $\{x : \sup_n f_n(x)>a\}$, and $\{f_n>a\}$ to mean $\{x : f_n(x)>a\}$. ${}\qquad{}$ – Michael Hardy Sep 20 '14 at 22:39

Your book seems to use the widespread notation $\{ f > a \} := \{ x : f(x) > a \}$. Hence the last identity of your post.