Proof involving limit superior/inferior of sequences of sets

**ben_pcc** · October 4th 2008, 04:19 PM

Hi.

I'm a mechanical engineering student in a real analysis class (yeah, I'm a little cocky).

As you can imagine I'm having difficulties, but there's this one thing that I'm stopped cold on. I need to prove that:

$\bigcup_{n = 1}^{\infty} \left(\bigcap_{k=n}^{\infty} A_k\right) \subset \bigcap_{n = 1}^{\infty} \left(\bigcup_{k=n}^{\infty} A_k\right)$

I get the intuitive meaning of this (I think so anyway) but I can't find a way to prove it. Especially the one way mentality (ie, it's not generally a proper subset). I've tried distributive laws, writing it out as English, and of course making up BS hoping to get a result that I can refine later. Waste of my precious Saturday so far.

To answer your question: yes, I registered just to ask. Sorry. Thanks in advance!

**ThePerfectHacker** · October 4th 2008, 04:34 PM

Originally Posted by ben_pcc

(yeah, I'm a little cocky)

You think you are? You never met me.

Remember: $A\subseteq B$ if and only if for any $a\in A \implies a\in B$ .

$\bigcup_{n = 1}^{\infty} \left(\bigcap_{k=n}^{\infty} A_k\right) \subseteq \bigcap_{n = 1}^{\infty} \left(\bigcup_{k=n}^{\infty} A_k\right)$

Let $x \in \bigcup_{n = 1}^{\infty} \left(\bigcap_{k=n}^{\infty} A_k\right)$ then $x \in \bigcap_{k=N}^{\infty} A_k$ for some $N\geq 1$ .
Thus, $x\in A_N, A_{N+1}, A_{N+2}, ...$ i.e. $x\in A_n$ for $n\geq N$ . [1]

To show that, $x\in \bigcap_{n = 1}^{\infty} \left(\bigcup_{k=n}^{\infty} A_k\right)$ it remains to show $x \in \bigcup_{k=n}^{\infty} A_k$ for all $n\geq 1$ . [2]

To prove [2] use the result established in [1].

and of course making up BS hoping to get a result that I can refine later.

That only works in English class when you write an essay.

**ben_pcc** · October 4th 2008, 04:57 PM

Spaciba tvarishch ThePerfectHacker!

That's too simple, I just can't think that way yet, haven't seen enough examples I guess.

That only works in English class when you write an essay.

Right, not in math, but it works in engineering too. Part of my graduate research involves finding solutions to fluid mechanics PDEs...

...by glaring over numeric results. Thanks for that tip though.

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