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I've encountered a problem where I have to show something about the measure of sets when the first set I'm asked about is a set containing infinite elements from infinite sets

$$B = {\bigcap_{k=1}^{\infty}} {\bigcup_{k=n}^{\infty}}An$$ $$E_k = {\bigcup_{k=n}^{\infty}}An$$

B is the desired set, as it takes infinite unions of $A_n$, each union is smaller as we proceed, and intersects them. This far is explained in the answer I've seen, but then it says that $E_1 \subseteq E_2$, but isn't this the other way around? $E_1$ is the union of all $A_n$ sets, while $E_2$ is the union of all $A_n$ beside $A_1$, is it not?

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    Yes, it's the other way around, $E_1\supseteq E_2.$ – 2017-02-02
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    also, I guess the k=n in the union, suppose to be n=k? – 2017-02-02
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    Just a note, the word **group** (see https://en.wikipedia.org/wiki/Group_(mathematics)) has a very specific meaning in mathematics, and it cannot be used interchangeably with **set**. Your question has nothing to do with *group theory*, it's all about sets. – 2017-02-02
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    @5xum you are right, sry. I don't learn in English so I try to translate as best as I can and sometimes I forget that set is the proper word to use – 2017-02-02
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    Oops, I didn't notice that. Yes, it should be $n=k$ to make sense. And the first equation should probably be $$B=\bigcap_{k=1}^\infty\bigcup_{n=k}^\infty A_n$$ if it's supposed to be $\limsup A_n.$ – 2017-02-02
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    I corrected my previous comment: lim sup, not lim inf. The elements of $B$ are things that belong to infinitely many of the $A_n.$ – 2017-02-02

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