I want to share my thought process and understanding of limit superior and inferior. If there are any flaws in my thought process at any step, please point that out. I will use $D(x_n)$ to mean the derived set (set of all limit points) of the sequence $(x_n)$. Note that, $D(x_n) \subseteq \mathbb{R}$
I define limit point of a sequence $(x_n)$ as $l\in \mathbb{R}$ such that $\forall \epsilon >0, x_n \in (l-\epsilon, l+ \epsilon)$ for infinitely many values of $n$.
If a sequence $(x_n)$ is bounded then $\limsup x_n$ and $\liminf x_n$ are simply the greatest and the least limit points of $(x_n)$ respectively.
Hence, $\limsup x_n = \max D(x_n)$ and $\liminf x_n = \min D(x_n)$
However, my textbook doesn't talk about unbounded sequence. So I looked it up and understood this– If $(x_n)$ is unbounded then $\max D(x_n)$ or $\min D(x_n)$ may or may not always exist. However we can give them a meaning so they always make sense and we may define them as:
- If $(x_n)$ is unbounded above then $\limsup x_n =+∞$
- If $(x_n)$ is unbounded below then $\liminf x_n=-∞$
However these definitions don't feel complete, for instance what about $\liminf x_n$ when $x_n$ is bounded below and unbounded above? How do I improve it, possibly in simple English without using complicated notations?
Edit: I suppose whether or not $\max D(x_n)$ or $\min D(x_n)$ exist is irrelevant. Since even if they exist then $\limsup x_n=+∞$ if $x_n$ is unbounded above, for instance.
Now, let me list some examples to check if I understand them well enough.
- If $$(x_n) = \begin{cases} 2+ \dfrac{1}{n}, \, n \text{ is even.} \\ \\ n, \, n \text{ is odd.} \end{cases}$$
Since $(x_n)$ is unbounded above AND bounded below with a least limit point of $2$ then $\limsup x_n = +∞$ and $\liminf x_n= 2$
- If $x_n= n$ then since $(x_n)$ is unbounded above AND bounded below but without a least limit point therefore, $\limsup = +∞$ and I guess $\liminf = +∞?$
Is my understanding correct? If there any corrections or improvements, please let me know.