I am required to prove that $x\mapsto\int_x^{x+1/n} n f(y)dy$ converges in the $L^1$ sense to $f$, knowing that $f\in L^1$.
My current attempt is: after a variable change, I've rewritten $x\mapsto\int_x^{x+1/n} n f(y)dy$ as $x\mapsto\int_0^{1} f(x+y/n)dy$, which enables me to verify that indeed $x\mapsto\int_x^{x+1/n} n f(y)dy$ converges pointwise to $f$. After that, I've tried to find a dominating function in $L^1$ for $x\mapsto\int_x^{x+1/n} n f(y)dy$ in order to use the Dominated Convergence Theorem, but I didn't succeed in finding such a function.
Any help is appreciated.