LimSup and LimInf Homework: Find (0,1)

[demZ]

Homework Statement

For $$n \in N$$, let A_n be (0,1+1/n) if n is even and (-1/n, 1) if n is odd.
Find Lim Sup A_n and Lim Inf A_n

Homework Equations

The Attempt at a Solution

Basically, I got Lim Sup = Lim Inf = (0,1). I'm not sure it's correct as I always get confused when I need to calculate UA_n. My usual approach is to draw the first elements of A_n, but I guess it's not a useful method when dealing with more complex A_n.

 

[micromass]

Hmm, using the definitions is annoying, I always use the following criterium

$$\liminf{A_n}=\{x~\vert~\exists n_0:~\forall n>n_0:~x\in A_n\}=\{x~\vert~x\in A_n~\text{eventually}\}$$

Using this, we see that the liminf of our sequence is (0,1).

For the limsup:

$$\liminf{A_n}=\{x~\vert~\forall n:~\exists m>n:~x\in A_m\}=\{x~\vert~x\in A_n~\text{for infinitely many n}\}$$

Using this, we see that the limsup of the sequence is (0,2).

 

[demZ]

Thanks for the quick reply! Can you provide the details? I'm not fully understand how to solve this kind of problems. The problem is fairly easy, but what is the correct approach/method when dealing with complex sequences?

Also, why $$\limsup{A_n}$$ is not (0, 1.5)?

 

[micromass]

Oh no, I was incorrect. The limsup is (0,1], not (0,2). 1 is in there because 1 occurs infinitely often in the sequence An (because 1 is in An whenever n is even).

So you are completely correct, except for the point 1. Maybe you should recheck your calculations for this point?