# Determine liminf$A_{n}$ and limsup$A_{n}$ for the $A_{n}$ of the following sets

$$A_{n} = \begin{cases} (\frac{1}{n}-2 , 1) \ \ \ \ \ \text{if} \ n \ \text{is odd} \\ (0,3 + \frac{1}{n}) \ \ \text{if} \ n \ \text{is even} \end{cases}$$

This question is on a past measure theory final and I think I have the right answer but would just like a second opinion.

I have:

• limsup($$\frac{1}{n}-2,1)$$= ($$-1,1$$)
• liminf($$\frac{1}{n}-2,1)$$=($$-2,1$$)
• limsup($$0,3 + \frac{1}{n})$$=($$0,3.5$$)
• liminf($$0,3 + \frac{1}{n})$$=($$0,3$$)

I was also wondering if the way I formatted my solutions would be acceptable on an exam? Thank you in advance.

• You have two too many answers. The sequence $A_n$ is one object, it doesn't break into separate answers for the even and odd cases. Also, in limsup and liminf, you are taking $n$ to infinity. That changes your answers, right? Commented Aug 1, 2020 at 16:51

I am afraid you are on the wrong track. The question seems to be about the whole sequence $$A_n$$, and you have separately evaluated two subsequences of it (the odd and the even members). Whatever the result, it must have a single $$\lim\sup_{n\in\mathbb N}A_n$$ and a single $$\lim\inf_{n\in\mathbb N}A_n$$.

To solve the problem: recall that $$\lim\sup_{n\in\mathbb N}A_n$$ is the set of all elements contained in infinitely many $$A_n$$'s. This will turn out to be $$(-2,3]$$. Similarly, $$\lim\inf_{n\in\mathbb N}A_n$$ is the set of those elements contained in all but finitely many of $$A_n$$'s, and it is easy to see that this set is $$(0,1)$$.

• This was really helpful, thank you very much!
– Jack
Commented Aug 1, 2020 at 17:01
• @Jack I did make a mistake, now corrected. $\lim\sup$ is $(-2,3]$, not $(-2,3)$. (Why?)
– user700480
Commented Aug 1, 2020 at 17:09

We have $$\limsup_{n\to\infty} 3+1/n = 3$$, $$\liminf_{n\to\infty} 1/n- 2 = -2$$, $$\liminf_{n\to\infty} 0 =0$$, $$\limsup_{n\to\infty} 1=1$$. So in summary

• $$\limsup_{n\to\infty} A_n = [0,3]$$
• $$\liminf_{n\to\infty} A_n = [-2,1]$$
• What definition of $\lim\sup$ and $\lim\inf$ are you using?
– user700480
Commented Aug 1, 2020 at 17:00
• $\liminf_{n\to\infty} a_n = \lim_{n\to\infty}\left(\inf_{k\geq n}a_k\right)$; similarly for limsup
– user813139
Commented Aug 1, 2020 at 17:00
• – user700480
Commented Aug 1, 2020 at 17:03