Limit of series of Heaviside step functions


I couldn’t find a better title, because it is a very specific limit that I want to show: $ $ f_b(y)=\frac{2}{b}\sum\limits_{k=1}^\infty\theta\left(y-\frac{k\pi}{b}\right) \rightarrow f(y)=\frac{2}{\pi}y,\;b\rightarrow\infty$ $ I thought, that it would be helpful to use this definition of the Heaviside step function: $ $ \theta=\lim\limits_{\epsilon\rightarrow0}\frac{1}{\pi}\left[\arctan(x/\epsilon)+\frac{\pi}{2}\right]$ $ Could it be some kind of Fourier series I have to work with?

Thanks for reading!