Let $\{x_1,x_2,x_3,...\}$ be an enumeration of the countable set $\mathbb{Q}\cap [0,1]$.
For $\varepsilon>0$ let
$U_{\varepsilon}=\bigcup_{n=1}^{\infty}\big( x_n-\frac{\varepsilon}{2^{n+1}} ,x_n+\frac{\varepsilon}{2^{n+1}} \big), \quad V_{\varepsilon}=U_{\varepsilon}\cap (0,1).$
Show that $V_{\varepsilon}$ is open and dense in $[0,1]$ and that $0<\lambda(V_{\varepsilon})\leq \varepsilon$.
I don't know how to start. Thanks for any help