I have the following function $u=\sum\_{k=1}^\infty \frac{(1)^k}{k}1\_{[k,k+1)}$ 

I want to show that this is Borel measurable. My idea is to show $\{u \geq a\} \in \mathcal{B}(\mathbb{R})$ according to my textbook. For a=1/2 I get $\emptyset \in \mathcal{B}(\mathbb{R})$. And for a=-1 I get $\mathbb{R} \in \mathcal{B}(\mathbb{R})$

But I don't how to show mathematically for other values of a. 

Any hint would be appreciated