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This is part of a bigger problem I'm working on.
To construct a a decreasing sequence of sets, $A_{n}\supseteq A_{n+1}$, I did the following: Let $B=\cup_{n=1}^{\infty} B_n$ and set $A_n= B\setminus (B_1\cup\ldots\cup B_n).$ Is this right? Do the $B_j's$ have to be disjoint?

How do I construct the increasing counterpart?

thanks.

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If you have a decreasing sequence, taking complement usually works for getting an increasing sequence. –  user21436 Jan 14 '12 at 19:26

1 Answer 1

To have that $A_n\subseteq A_{n-1}$ you don't have to have the $B_n$'s disjoint.

We have that that $x\in A_n$ if and only if $x\in B_k$ for some $k>n$. This means that if $x\in A_{n+1}$ then it is in $B_k$ for some $k>n+1>n$ therefore $x\in B_k$. If you want an increasing sequence $C_n\subseteq C_{n+1}$ we can simply take $C_n = B_1\cup\ldots\cup B_n$.

If the sets $B_n$ are not disjoint, or strictly increasing in $\subseteq$ then it is fairly possible to have $C_k=C_{k+1}$ for some $k$ (and similarly with the $A_k$'s).

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Thanks. how would I construct the increasing counterpart? –  jojo Jan 14 '12 at 19:43
    
@jojo: Did you post this after the edit? –  Asaf Karagila Jan 14 '12 at 19:43
    
before the edit. –  jojo Jan 14 '12 at 19:52
    
@jojo: Did the edit answer your question? –  Asaf Karagila Jan 14 '12 at 20:12
    
Yes! Thank you again. –  jojo Jan 14 '12 at 20:18

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