real analysis - Does ${sin (nx)}_1^infty$ converge in the $L^1$ norm on $[0,2pi]$?

This is a homework question from a problem set in an undergraduate-level real analysis course (coming from merely an intro to analysis course) about $L^p$ spaces.

Show that $\{\sin (nx)\}_{n=1}^\infty$ converges in the $L^1$ norm on $[0,2\pi]$

I showed that, for $f_n(x)=\sin(nx)$, the sequence of norms $\lVert f_n \rVert$ converges, but apparently I was supposed to show that $\lVert f-f_n\rVert\to0$, which I'm not really sure how to do. I'm probably missing something relatively simple, but I would appreciate the help.