# Math Help - Measure theory

1. ## Measure theory

Show that (a,b) = $\bigcup_{n\in N}$( $a_{n},b_{n}$) , where $a,b\in\mathbb{R}$ , { $a_{n}$} is a decreasing sequence of rational numbers with $a_{n}\rightarrow a$ , { $b_{n}$} is an increasing sequence of rational numbers with $b_{n}\rightarrow b$ .

2. Suppose x is in (a, b). That is, that a< x< b. Show that, for some n, $a_n< x< b_n$. That should be easy. Take $\epsilon= x-a$. Since $a_n \to a$, there exist n such that $|a- a_n|< \delta$ so $a. Since $b_n\to b$, there exist m such that $|b-b_n|< \delta$ so $b< b_n< b$.

It should be clear that is x< a, then x is in NONE of $(a_n, b_n)$ and so not in their union. Similarly for x> b. The important part is x= a and x= b. You need to show that $a< a_n$ and $b_n< b$ for all n.