What does $\liminf_{n\to \infty} f_n$ and $\limsup_{n\to \infty} f_n$ mean?

Question

I have searched the internet, including this website, and I have found some answers, but none which I understand what actually mean.

What is $\liminf_{n\to \infty} f_n$? Is it a single value? Is it a sequence of values? Is it a function?

Is $\liminf f_n$ equivalent to any of these:

1. $\liminf_{n\to \infty} {f_n}$ = {$g(x)$ : $g(x)$ is the smallest function value in the sequence $f_n$ at each $x$ for any $n$}
2. $\liminf f_n$ = a, such that $a$ is the smallest function value for any $n$ and $x$?
3. $\liminf f_n$ = $f_n$ such that value of the norm $\|f_n\|$, is the smallest of any of the functions in {$f_n$}
4. Any other?

Please don't just label this question a duplicate. I have read the similar topics on this website, but I don't either (i) understand the answers given, or (ii) didn't seem relevant.

Thank you for your time.

Kind regards,

Marius

Answer 1

First let's suppose that you have a sequence $\{r_n\}_{n=1}^{\infty} \subset \Bbb{R}$. Then $$ \liminf_{n \to \infty} r_n = \lim_{n \to \infty} \inf\{ r_k : k \geq n\}. $$ and $$ \liminf_{n \to \infty} r_n = \lim_{n \to \infty} \sup\{ r_k : k \geq n\}. $$ If we allow for the values $\pm \infty$, then the $\liminf$ and $\limsup$ always exist since they are limits of non-decreasing and non-increasing sequence, respectively.

Now, if $\{f_n\}_{n=1}^{\infty}$ is a sequence of functions having a common domain, then for a fixed value $x$ in the domain of the functions, $\{f_n(x)\}_{n=1}^{\infty}$ is a sequence of real numbers (assuming that the codomain of the $f_n$ is $\Bbb{R}$). So, $\liminf_{n\to \infty} f_n$ yields a function $f$ where $f(x) = \liminf_{n\to \infty} f_n(x)$; note that $\liminf_{n\to \infty} f_n(x)$ is now just a $\liminf$ of a sequence of numbers $\{f_n(x)\}$. A similar story holds for $\limsup_{n \to \infty} f_n$. Hope this helps.

Answer 2

Any answer to this really has to have two parts.

• What is it? I like to think of $\limsup_{n \to \infty} f_n$ as the "eventual least upper bound" of the sequence $(f_n)_{n = 1}^\infty$, and correspondingly, $\liminf$ as the "eventual greatest lower bound". What this means is that for any $m \geq 1$ we can consider the least upper bound of just those terms of $(f_n)$ with $n \geq m$; correspondingly, the greatest lower bound:

  $s_m = \sup\{f_n \mid m \geq n\} \qquad i_m = \inf\{f_n \mid m \geq n\}$

  These new sequences are more well-behaved than $(f_n)$ itself; in particular, as $m$ increases, $s_m$ is decreasing and $i_m$ is increasing, whereas $f_n$ is probably neither. Therefore, the limit of $(s_m)$ is its infimum, and the limit of $(i_m)$ is its supremum, and these, by definition, are the $\limsup$ and $\liminf$ of $(f_n)$ itself:

  $\begin{align} \liminf_{n \to \infty} f_n &= \lim_{m \to \infty} s_m = \inf \{s_m\} = \inf \bigl\{ \sup\{f_n \mid n \geq m\} \bigr\} \\ \limsup_{n \to \infty} f_n &= \lim_{m \to \infty} i_m = \sup \{i_m\} = \sup \bigl\{ \inf\{f_n \mid n \geq m\} \bigr\}. \end{align}$

• Why use it? There's a reason these definitions only appear in theoretical analysis and not in basic calculus, despite the fact that they can be formulated in terms of limits: they are only useful in discussing questions of the existence of limits. Basically, they are tools for establishing foundations. As such, the concept of $\limsup$ (and $\liminf$) is more basic than the concept of a limit, at least when your axiomatic development of real analysis starts from the following unique characterization of $\mathbb{R}$:

  $\mathbb{R}$ is the unique ordered field in which every set with an upper bound has a least upper bound.

  This establishes $\sup$ as the inherent limit concept in $\mathbb{R}$. A little work shows that for any increasing sequence $(f_n)$, its supremum $\sup \{f_n\}$ (as a set alone, not a sequence) satisfies the epsilon-delta definition of a limit, and likewise for any decreasing sequence its infimum also does. But this is not true for non-monotone sequences; for example; the oscillating sequence

  $f_n = (-1)^{n - 1} \dfrac{n + 1}{n}, \quad n \geq 1$

  first, does not have a limit; and second, does not even appear to "follow" its supremum, which is $2$, or infimum, which is $-3/2$. In fact, its primary behavior is to oscillate between "envelopes" that tend, as $n \to \infty$, to $\pm 1$. In this case, though, those envelopes are approached from the "outside", so you need to discard excessively large initial terms to see the limiting behavior. But this is exactly what the above definitions of $s_m$ and $i_m$ do: you take the tightest upper and lower bounds after the sequence "settles down" a bit, and tighten them even more by making "a bit" a moving target.

  The end result is that the sequence will seem, in part, to be "attracted" to $\limsup f_n$ and in part to $\liminf f_n$, and while it may have other points of attraction, they will be in between. So if $\limsup f_n = \liminf f_n$, we know that $\lim_{n \to \infty} f_n$ exists and is this common value, and if not, we also have a partial quantification of in what way $(f_n)$ fails to have a limit as $n \to \infty$.

So like I said, the limits superior and inferior are foundational. In basic calculus you don't care about this kind of deep investigation, but only about making some simple intuition somewhat precise, and mostly about doing computations. The concept of an actual limit is really defined in terms of $\limsup$ and $\liminf$, and there are definitely important theorems for which these are the relevant concept of limit. If you take nothing away from this, take this away: increasing sequences are the best kind of sequence, and their limit is their supremum. That's what $\limsup$ means.