I am unable to ascertain my result of the following problem. Can anyone help me out? Problem: Let $X=(0,1)$ and $U_n=(1/n,1), n\geq 2$. Does a Lebesgue number exist for this cover?
Solution: Consider $A=(0,1/n), n\geq N$. Now $\nexists U_N$ which can cover $A$.Hence no Lebesgue number exist.
Is my arguement valid? Thanks for help.
As I have said, it is enough to show that for any $\delta > 0$ there exists some $A\subseteq X$ s.t. $\text{diam}(A) < \delta$ but $A \not\subseteq U_n$ for any $n \in \mathbb{N}$.
So let $\delta > 0$ be arbitrary. Then consider $A := (0, \frac{1}{n})$ with $n \in \mathbb{N}$ s.t. $\frac{1}{n} < \delta$. Then we have $\text{diam}(A) = \frac{1}{n} < \delta$. Now we just need to show that $A \not\subseteq U_n$ for any $n \in \mathbb{N}$. To show this, we need to show that for any $n \in \mathbb{N}$ there exists some $x \in A$ s.t. $x \not\in U_n$.
So let $n \in \mathbb{N}$ be arbitrary. Then choose $x := \frac{1}{2n}$. Then we have $x \not\in U_n$. This concludes the proof.