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Thread: Measure theory

  1. #1
    Member SENTINEL4's Avatar
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    Measure theory

    Show that (a,b) = $\displaystyle \bigcup_{n\in N}$($\displaystyle a_{n},b_{n}$) , where $\displaystyle a,b\in\mathbb{R}$ , {$\displaystyle a_{n}$} is a decreasing sequence of rational numbers with $\displaystyle a_{n}\rightarrow a$ , {$\displaystyle b_{n}$} is an increasing sequence of rational numbers with $\displaystyle b_{n}\rightarrow b$ .
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  2. #2
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    Suppose x is in (a, b). That is, that a< x< b. Show that, for some n, $\displaystyle a_n< x< b_n$. That should be easy. Take $\displaystyle \epsilon= x-a$. Since $\displaystyle a_n \to a$, there exist n such that $\displaystyle |a- a_n|< \delta$ so $\displaystyle a<a_n< x$. Since $\displaystyle b_n\to b$, there exist m such that $\displaystyle |b-b_n|< \delta$ so $\displaystyle b< b_n< b$.

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