I started reading Probability & Measure Theory by Robert Ash and Catherine Doleans-Dade and got stuck at the second exercise in the very first section.
Define sets of real numbers as follows. Let $A_n=(-1/n, 1]$ if $n$ is odd, and $A_n=(-1, 1/n]$ if $n$ is even. Find $\lim\sup_n A_n$ and $\lim\inf_n A_n$ (limit superior and limit inferior).
I got the correct solution for $\lim\sup_n A_n$ (it is $(-1, 1]$), but it seems that I'm making some mistake(s) for $\lim\inf_n$.
Here's my solution: $$ \begin{matrix} A_1=(-1, 1] & A_2 = (-1, 1/2] \\ A_3 = (-1/3, 1] & A_4 = (-1, 1/4] \\ A_5 = (-1/5, 1] &A_6 = (-1, 1/6] \\ \vdots & \vdots \\ A_{2k+1} = (\frac{-1}{(2k+1)}, 1] & A_{2k}=(-1, \frac 1 {2k}] \\ \vdots & \vdots \\ \end{matrix} $$ We can see that $A_1 \supset A_3 \supset A_5 \supset \ldots$ and $A_2 \supset A_4 \supset A_6 \supset \ldots$.
We can also see that $\lim_{k \rightarrow \infty}A_{2k} = (-1, 0]$ and $\lim_{k \rightarrow \infty}A_{2k+1}=(0, 1]$.
From there, I concluded that $$\bigcap_{even\ k \geq n}^{\infty}A_k=(-1, 0]$$ and $$\bigcap_{odd\ k \geq n}^{\infty}A_{k}=(0, 1]$$ for each $n$, and got the following result:
$$ \bigcap_{k=n}^{\infty}A_n=\left(\bigcap_{even\ k \geq n}^{\infty}A_{k}\right)\cap\left(\bigcap_{odd\ k \geq n}^{\infty}A_{k}\right) = (-1, 0] \cap (0, 1] = \emptyset $$
So: $$ \lim\inf_n A_n = \bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k=\bigcup_{n=1}^{\infty}\emptyset=\emptyset $$
However, the correct answer listed at the end of the book is $\{0\}$.
Would anyone be so kind to check my solution?
$\lim A_{2k+1}=[0,1]$. Note that $0 \in A_{2k+1}$ for each $k$.
[The intervals are open on the left but that doesn't mean that you have to leave out $0$ in the limit. You have to include it because $0 \in A_{2k+1}$ for each $k$].