real analysis - The definitions of limit infimum and limit supremum

I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit of a sequence of real numbers as $\lim_{n \to \infty}x_n=x$ if for each $0 <\epsilon$ there is a natural number $N$ such that it is $|x-x_n| < \epsilon$ for $n > N$.

Then the limit infimum and supremum are defined as $\liminf_n x_n = \lim_{n \to \infty} \inf_{k\geq n} x_k$ and $\limsup_n x_n = \lim_{n \to \infty} \sup_{k\geq n} x_k$.

1. My first question is about the interpretation of the limit infimum and limit supremum definitions. It looks to me like they are defined as the limits of sequences as well. For example, for the infimum, one can define a sequence $a_n = \inf_{k \geq n}x_k$ where $x_n$ was already a sequence of real numbers. Then the limit infimum is the limit of the sequence $a_n$ , $\lim_{n \to \infty}a_n$, so the infimums of the subsequences $x_k$ where $k \geq n$ are converging if the limit exists. It is the similar for the supremum. Is my interpretation correct here?




2. It is said the limit infimum and supremum do always exist, though they can be infinite. I did not understand this; why it is so?




3. In the book it is stated that the limit $\lim_{n \to \infty} x_n$ exists if and only if it is $\liminf_n x_n = \limsup_n x_n$. I could not prove this to myself. How can it be shown that this statement is true?

Thanks in advance.

Answer

These are answers to the secondary questions
in the comments after my answer to the original question;
I cannot put the answers in comments,
because the size of a comment is limited.

The second question.
Let $(x_n)$ be a sequence taking values in the set $\{0,1,2,\ldots,9\}$,

and $a:=\liminf_n x_n$.
If $0$ appears infinitely many times in the sequence, then $a=0$.
If $0$ appears only finitely many times, but $1$ appears infinitely often, then $a=1$.
And so on.
This is a slick answer.
It is entirely another matter to answer the question with a definite value
for a particular given sequence, such as the sequence of digits after the comma
in the decimal expansion of $\pi$.
Well, if you can prove that the digit $0$ appears infinitely many times
in the decimal expansion of $\pi$, then you have $a=0$, and so on;

in this case it takes a serious theoretical effort to determine the $\liminf$.

As for the third question,
let $(x_n)$ be a sequence of real numbers
such that $a:=\liminf_n x_n$ is a real number (that is, $-\inftySince the sequence of infima $a_n:=\inf_{k\geq n} x_n$ is increasing
(meaning that $m\leq n$ implies $a_m\leq a_n$ -- I hate "nondecreasing"),
the limit $a=\lim_{n\to\infty} a_n$ is actually the supremum: $a=\sup_n a_n$.
$\quad$Let $a_1\leq a$ and $a'We claim that there exists $m$ such that $x_n\geq a'$ for every $n\geq m$:
since $a=\sup_n a_n$, there exists $m$ such that $a_m\geq a'$;

but then $x_n\geq a_m\geq a'$ for every $n\geq m$.
$\quad$Now let $a_1>a$, and set $a':=\tfrac{1}{2}(a+a_1)$; we have $aIn this case we claim that given any $m$
there exists $n\geq m$ such that $x_nsince $a_m=\inf_{k\geq m} x_k\leq a$\quad$We have proved that $a$ is the largest of all real numbers $a_1$
with the property that
for every $a'