Is the function $f(x, y) = \sum\limits_{n=1}^{\infty} \frac{[xn]}{1 + e^{n + [y]}}$ Borel on $\mathbb{R}^2$.
I tried to estimate this function first, but I was only able to estimate it from above, and for $0 \leq [y]$. I got that $$\sum\limits_{n=1}^{\infty} \frac{[xn]}{1 + e^{n + [y]}} \leq \sum\limits_{n=1}^{\infty} \frac{xn}{e^n} \leq x \sum\limits_{n=1}^{\infty} \frac{n}{e^n}$$ And $\sum\limits_{n=1}^{\infty} \frac{n}{e^n}$ converges according to D'Alembert criterion. But I don’t know what to do next, please tell me how to approach this example