Tuesday, January 25, 2022

Differences between sup / inf and lim sup / lim inf

Students who took the Calculus class like me will learn about two very similar concepts: the sup / inf of a sequence and the lim sup / lim inf of a sequence. While these two terms are very similar there are some fundamental differences between them. My friend asked my why they are not equal so I decided to write this blog post. Let's start by giving the formal definitions, then I will try to give an intuitive explanation as to why they are different.


The definitions for the supremum and the infimum use the terms upper bound and lower bound. Let S be a nonempty subset of $\mathbb{R}$, 

$u$ is called an upper bound of S if $ \forall x \in S $ $ u \ge x $. 

The set of all upper bounds is denoted by $ U_S $. And similarly, 

$l$ is called an lower bound of S if $ \forall x \in S $ $ l \le x $. 

The set of all lower bounds is denoted by $ L_S $.

For example if you have the set [0,1] then 10 is an upper bound since 10 is bigger than any number between 0 and 1. 1 is also an upper bound since it is bigger or equal to any number between 0 and 1.


A real number M is called the least upper bound of S or the supremum of S or just simply sup S if:

$M \in U_S$ and $\forall u \in U_S$ $M \le u$

A real number m is called the greatest lower bound of S or the infimum of S or just simply inf S if:

$m \in L_S$ and $\forall l \in L_S$ $m \ge l$ 

Like in the above example both 10 and 1 are upper bounds, but only 1 is the supremum since it is the smallest upper bound. 

Let ${x_n}$ be a sequence in $\mathbb{R}$. We define,

 \[ \text{lim sup }x_n := lim_{N \rightarrow \infty} sup\{x_n : n>N\} \]

 \[ \text{lim inf }x_n := lim_{N \rightarrow \infty} inf\{x_n : n>N\} \]

Intuitively  you can think this as dismissing the first terms of the sequence and just looking at the supremum in the most right side of the graph. The difference between them is that lim sup $x_n$ can be smaller than the supremum. The best example that I can give to this is:

\[ x_n := \begin{cases} 100 & n=1 \\ 1 & n \ne 1 \end{cases} \]


sup $x_n=100$ while lim sup $x_n$=1 

Another difference between them is that if $x_n$ converges, then lim sup $x_n$ = lim inf $x_n$. Of course we can prove this formally but that proof already exists in textbooks. I want to give an intuitive reason why for this. Think of

\[ y_n :=  \begin{cases} \frac{1}{n} & n \text{ is even} \\ -\frac{1}{n} & n \text{ is odd} \end{cases} \]


sup $y_n=1$, inf $y_n=-1$ but lim sup $x_n$ = lim inf $x_n$= lim $x_n$ =0. As you can see from the graph, when we look at the right side, all of the values get invisibly close to zero, so among these numbers the supremum and infimum is zero.

Of course my intuitive explanation might make mathematicians want to pull their hair out but in my opinion even if they help one person, it is good to have explanations like these.

Note: The graphs were drawn by me so they are not precise. You get the point. If this post helped you understand something feel free to comment, it would certainly make my day.



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