I am reviewing for a final exam and it seems as though I am still not understanding $\limsup$, $\liminf$, $\min$ and $\max$. I wanted to check my understanding here if you don’t mind. Basically, I have a 2 part question.
If I understand this correctly, $\limsup$ and $\liminf$ are basically found by taking an arbitrary sub sequence and analyzing it as the index goes to infinity, yes? With that said, if it is bounded (containing brackets) then clears contain the $\liminf$ and sup respectively, yes? Moreover, I know the $\min$ and $\max$ are the lowest and highest values in the set, but how come they sometimes don’t exist?
Here is a sample problem I am reviewing:
$X=\left\lbrace x\in[0,1] | x \text{ is irrational}\right\rbrace$
I know that since the set is bounded between $0$ and $1$ that my $\liminf$ is $0$ and my $\limsup$ is $1$, correct? My $\min$ and $\max$ should also be $0$ and $1$, yes?
First of all $\sup,\ \inf,\ \max,\ \min$ are defined for sets.
$\sup A$ is a number $a$ s. t. $\forall \alpha \in A,\ \alpha \leq a$ AND there is no number $b$ s. t. $\forall \alpha \in A, \alpha \leq b \leq a$. So it is the smallest number, that bounds $A$. For example $\sup (0;1) = 1$, and $\sup [0;\infty)$ doesn`t exist.
$\inf$ defined similary, but with $\geq$ instead of $\leq$.
$\max$ and $\min$ are similar but should be contained in the set. For example for $(0;1)$ neither $\min$ nor $\max$ exists, $\min [0,1) = 0$, $\max$ not exists, etc.
$\lim \sup$ has a different nature. It is defined for functions or sequences. It basically supremum as function tends to some point. And makes sense for oscillating functions for example. I.e. there is no $\lim\limits_{x \rightarrow \infty}\sin x$, but $\lim\limits_{x \rightarrow \infty}\sup \sin x = 1$.
So for your set $\sup, \inf$ are $0, 1$ correspondingly. $min, max$ do not exist. And $\lim \sup, \lim\inf$ make no sense.