1. ## sequence counterexample

Give a counterexample:

lim sup ($\displaystyle a_n$ + $\displaystyle b_n$) = lim sup $\displaystyle a_n$ + lim sup $\displaystyle b_n$

Give a proof or counter-example:

A nested sequence of bounded intervals must have a non-empty intersection.

2. Originally Posted by glasses123
Give a proof or counter-example:
A nested sequence of bounded intervals must have a non-empty intersection.
This is a perfect example of where different definitions give different results.
If we allow that $\displaystyle A_n = \left( {0,\frac{1}{n}} \right)$ to be a sequence of nested intervals you get one result.
But many insist that an interval is closed, in that case we get another result.