Let $C_1$ be the collection of intervals of the form $[\frac{k}{2^n},\frac{k+1}{2^n})$ where $n = 1, 2, 3, . . . and $ k ∈ Z , together with the empty set. a) Show that C_1 generates the Borel σ-algebra on R. b) Show clearly and explicitly that $C_1$ is a p-system.
I think for first part (a) my idea is the collection say $ C_2 =(\infty, b]$ where $ b \in R $ generated Borel sigma algebra. so I just get inclusion between them. Even for part (b) I have no idea.
For part (a), if you already know that intervals of the form $(-\infty,b]$ generate the Borel $\sigma$-algebra, then it's enough to show that $\sigma(C_1)$ contains all intervals of the form $(-\infty,b]$.
To show this, you can first prove that $\sigma(C_1)$ contains all intervals of the form $(-\infty,d)$ where $d$ is a dyadic rational number (i.e. of the form $\frac{k}{2^n}$ for integers $n\geq 1$ and $k$), then use the fact that the dyadic rational numbers are dense in $\mathbb{R}$ to write any $b\in\mathbb{R}$ as the limit of a decreasing sequence of dyadic rational numbers.
For part (b), if by p-system you mean what's often called a $\pi$-system, then you just have to show that $C_1$ is closed under intersections. You might find it helpful to prove that if two intervals of the form $[\frac{k}{2^n},\frac{k+1}{2^n})$ are not disjoint, then one is contained in the other.