I am trying to see why $\frac{\sin(nx)}{1+\vert x\vert^2}$ has no convergent subsequence in $L^1(\mathbb{R})$. This is an "optional" homework problem from my disorganized analysis professor which is worth no points, and which I wouldn't do anyway, because I'm terrible at analysis and have no clue how to deal with this.
To me it seems clear though that I want to be looking at the integral $\int\limits_\mathbb{R} \frac{\vert\sin(n_kx)-\sin(n_jx)\vert}{1+\vert(x)\vert^2}$ and finding some kind of constant, maybe in terms of when $n_k\neq n_j$ based on vague memories of undergrad PDE. Any hints?