# Set theoretic problem, lim sup and lim inf

1. Apr 27, 2012

### sunjin09

1. The problem statement, all variables and given/known data

Given a sequence of sets {An}, n=1,2,...,∞, the lim sup $S=\cap_{n=1}^\infty\cup^\infty_{k=n}A_k$, and the lim inf $I=\cup_{n=1}^\infty\cap^\infty_{k=n}A_k$, obviously $I\subset S$, find an expression for the set difference $S-I$

2. Relevant equations

$(A \cup B) \cap C = (A \cap C)\cup(B\cap C)\,$
$(A \cap B) \cup C = (A \cup C)\cap(B\cup C)\,$

3. The attempt at a solution
I don't know how to use the set algebra identities to simplify, I need to write S-I in terms of set differences of An. Are there any identities involving infinite union/intersection that might be of help? Thank you.

Last edited: Apr 27, 2012