Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have a question about whether the following two sets on the two sides are the same:

$$ \lim_{a \rightarrow \infty} \, \limsup_{n \rightarrow \infty} \, \{ x \in S \, | \, f_n(x) > a \} = \left\{ x \in S \, \Bigm| \, \lim_{n \rightarrow \infty} f_n(x) =\infty \right\}\quad ? $$

where $\{f_n\}$ is a sequence of real-valued functions defined on a set $S$. How can you explain it?

Thanks in advance!

share|cite|improve this question
Here is what I understand, on the left hand side, first take the limsup of a sequence of sets and the result is still a set, which is parameterized by a, and then let a goes to infinity. So the lhs is a set, isn't it? – Mary Dec 12 '10 at 18:46
You are talking about the limsup of a sequence of sets. How is that defined? – Tsuyoshi Ito Dec 12 '10 at 19:00… – Mary Dec 12 '10 at 19:03
Try with a sequence of functions that has an alternating component that gets larger in absolute value with $n$ but changes sign to see if this still holds. It shouldn't if I read the definition of limsup for sets correctly. – Raskolnikov Dec 12 '10 at 19:04
@ Raskolnikov: can you explain more? How about if {f_n} are all positive functions? I tend to think the two sets are the same, and don't see it make differences whether {f_n} are positive – Mary Dec 12 '10 at 19:10
up vote 4 down vote accepted

Consider a sequence of functions $f_n:\mathbb{R} \to \mathbb{R}$ defined by $f_n {(x)} = n$ if $x=0$ and $n$ is even, and $f_n {(x)} = 0$ otherwise. The right-hand side is obviously $\emptyset$, while the left-hand should be $\lbrace 0 \rbrace$.

EDIT: Answering the additional questions.

Suppose that $f_n{(x)}$ is an increasing sequence for each $x$. If $x$ belongs to the left set, then it must belong to the set $\lim \sup _{n \to \infty } \{ x \in S|f_n (x) > a\}$ for any $a > 0$ fixed. This means that for any $a>0$, there are infinitely many $n$ such that $f_n (x) > a$. Since $f_n (x)$ in increasing, it must converge to a positive number, or diverge to $\infty$. But $a>0$ is arbitrary, hence $f_n (x)$ must diverge to infinity. That is, $x$ belongs to the right set. Since the right set is contained in the left one, we conclude that both sets are equal.

EDIT: The fact that the right set is contained in the left one, is proved as follows. Suppose that $x$ belongs to the right set, and let $a>0$ be arbitrary but fixed. Then, for all sufficiently large $n$, $f_n {(x)} > a$. In particular, $x \in \lim \sup _{n \to \infty } \{ x \in S|f_n (x) > a\}$. Since this is true for any $a > 0$, $x$ belongs to the left set.

EDIT: The following point should be stressed. Denote by $E_a$ the set $\lim \sup _{n \to \infty } \{ x \in S|f_n (x) > a\}$. If $x \in E_a$, then $x \in E_{a'}$ for any $a' < a$. It follows that $\lim \sup _{a \to \infty} E_a = \lim \inf _{a \to \infty} E_a$; hence, by definition, the limit $\lim _{a \to \infty} E_a$ exists, and is equal to $\lim \sup _{a \to \infty} E_a = \lim \inf _{a \to \infty} E_a$. So, the left set in the question is indeed properly defined, and $x \in \lim _{a \to \infty} E_a$ means, in our case, that $x \in E_a$ for every $a$. Finally, note that always the $\lim \inf$ is a subset of the $\lim \sup$ (in analogy with the case of sequences of real numbers, where $\leq$ plays the role of $\subseteq$).

share|cite|improve this answer
@Shai:Thanks! Is it true that the left set always a super set of the right set? – Mary Dec 12 '10 at 19:21
Makes me wonder if replacing the lim in the right hand side with a limsup makes it true? I still think it's not, but I'm not sure. – Raskolnikov Dec 12 '10 at 19:24
@Mary: That's probably true, but I'll check this. – Shai Covo Dec 12 '10 at 19:28
@ Raskolnikov: are you talking about if it is true that the two sets are same, or if the left one is a superset of the right one? – Mary Dec 12 '10 at 19:29
Just if the sets are the same. – Raskolnikov Dec 12 '10 at 19:42

For question number two, construct the following sequence of $f_n(x)$

$$ f_n(x) = \left\{\begin{array}{ll} k+\epsilon_n, & \mbox{for $x=1/k$ with $k=1,\ldots,n$,} \\ \epsilon_n, & \mbox{otherwise.}

With the sequence of $\epsilon_n=1-\frac{1}{2^n}$.

If I did not make any mistake, the right-hand side should be empty, while the left-hand side contains $0$. (This example was constructed with the general set convergence in mind.)

share|cite|improve this answer
Thanks! your example is a wonder. How to understand the left-hand side contains 0? – Mary Dec 12 '10 at 20:42

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.