# How to construct an increasing sequence of sets.

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

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).