$g_n(x)=f(x+1/n)$ converges to $f$ in $L^1(\mathbb{R}$

Question

This is a question from my previous homework, which I didn't do and want a solution now since I am reviewing for my final.

"Suppose $f:\mathbb{R} \to \mathbb{R}$ is in $L^1(\mathbb{R})$. Prove that $g_n(x)=f(x+1/n)$ converges to $f$ in $L^1(\mathbb{R})$"

I could not do it before since it LOOKS obvious. Since the $g_n(x)$ LOOKS LIKE it converges pointwise to $f$ so it gives me such an impression. I am so stuck with this idea that I do not know how to approach properly.

Of course, converge almost everywhere does not implies convergence in $L^1$. So in a way my approach is wrong. Can you help?

Answer 1

It is a known fact that $$\lim_{t \to 0}\int|f(x+t)-f(x)|dx \to 0$$ for Lebesgue integrable functions.

This can be proved by approximation with simple functions.

So use this with $t_n=\frac{1}{n} \to 0$