Let $f\in L^1(\mathbb{R})$, and let $r_n$ be a sequence of numbers converging to $0$. I want to show that $f_n(x) = f(x+r_n)$ converges to $f$ in $L^{1}(\mathbb{R})$.
I think this problem can be solved by using the density of the step function. Choose a step function $\psi$ such that $\| f-\psi \|_{L^{1}} \leq \epsilon/3$, so we have $$ \left\|f_n-f\right\|_{L^1} \leq\left\|f\left(x+r_n\right)-\psi\left(x+r_n\right)\right\|_{L^1}+\left\|\psi\left(x+r_n\right)-\psi(x)\right\|_{L^1}+\|\psi-f\|_{L^1} $$ and hence $\left\|f_n-f\right\|_{L^1} \leq 2 \epsilon / 3+\left\|\psi\left(x+r_n\right)-\psi(x)\right\|_{L^1}$ We can see that since $\psi $ is a step function, so $\left\|\psi\left(x+r_n\right)-\psi(x)\right\|_{L^1}\leq 2\left|r_n\right| \rightarrow 0$, which finishes the proof.
My question is: for this question, are there any possibilities for me to prove it by using the dominated convergent theorem?