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I picked up this book called "Lebesgue Integration on Euclidean Space" but I cannot find a solutions manual for it anywhere. Problem 2 talks about limsup and liminf.

It goes like this: for a sequence of sets $A_1,A_2,...$, prove that

$$\cap_{j=1}^{\infty} \left( \cup_{k=j}^{\infty} A_k \right) = \{ x| x \in A_k \text{for infinitely many} \ \ k \}$$

My attempt at it:

If we look at the union of sets inside the brackets then we will have:

$$A_j \cup A_{j+1} \cup ...$$

and so we can rewrite LHS:

$$\cap_{j=1}^{\infty} \left( A_j \cup A_{j+1} \cup ... \right) = (A_1 \cup A_{2} \cup ...) \cap (A_2 \cup A_3 \cup ...) \cap ...$$

And so naturally due to intersection we will only have $x \in A_k$ for infinitely large $k$ in our final set, however, that is not what the problem asks to prove. It asks to prove that the final set will be composed of $x \in A_k$ for infinitely many $k$. So I do not think I have proved anything.

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    Does a sequence of sets in this case mean $A_1 \subset A_2 \subset A_3 ...$ ? What if one set from this sequence is $\emptyset$ – 2017-01-29
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    Is the index i used in the intersection supposed to be j? I don't see i used anywhere else – 2017-01-29

1 Answers 1

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Suppose $x \in A_i$ for infinitely many $i$. Then $x \in \bigcup_{k = j}^\infty A_k$ since we find an $i \geq k$ because there are infinitely many ones. So especially $x \in \bigcap_{j = 1}^\infty \bigcup_{k = j}^\infty A_k$. Conversly suppose $x \in \bigcap_{j = 1}^\infty \bigcup_{k = j}^\infty A_k$. Then $x \in \bigcup_{k = j}^\infty A_k$ for any $j \in \mathbb{N}$. Which means that $x \in A_k$ for some $k \leq j$ for all $j \in \mathbb{N}$. Hence $x \in A_k$ for infinitely many $k$.