Dominated Convergence of a shifted function

Question

I want to pass the limit into the following integral where $f$ is a $L^2(\mathbb{R})$ function. $$ lim_{y \to x} \int_\mathbb{R} |f(k)|^2|e^{ikx}-e^{iky}|^2 dk $$ I know that this can be done by dominated convergence but I seem to be unable to find the dominating function.

The goal would be to show that this integral is equal to 0.

Answer 1

I might be missing something but can't u just use $$|e^{ikx}-e^{iky}|\leq |e^{ikx}|+ |e^{iky}| \leq 1 +1$$ and then appeal to the $L^2$-property of $f$?