I want to show the following limit $$\lim_{\epsilon\downarrow0}\int_{1}^{y/\epsilon}(y+\epsilon x)^{-a}f(x)\;dx=y^{-a}\int_{1}^{\infty}f(x)\;dx,$$ assuming the integral on the RHS is finite, and $f$ is smooth an decreasing. The integral on the left can be written as
$$\int_{1}^{y/\epsilon}(y+\epsilon x)^{-a}f(x)\;dx=\int_{\mathbb{R}}(y+\epsilon x)^{-a}\chi_{[1,y/\epsilon]}(x)f(x)\;dx.$$
By dominated convergence it is enough to show $$\lim_{\epsilon\downarrow0} (y+\epsilon x)^{-a}\chi_{[1,y/\epsilon]}(x)\to y^{-a}\chi_{[1,\infty)}(x)$$
Intuitively, this limit follows. In an attempt to make it rigorous the only existing results have been of the type $\chi_{E_n}\to \chi_{E}$ when $\liminf_nE_n=E$, i.e. only for countable indexes. Does this result extend to the case I have?, thanks.