Showing that a sequence of random variables with $\mathcal{L}(X_{n})$ (I.e., law) uniform on $[-n,n]$ does not converge at all Let $(X_{n})_{n\geq 1}$ be a sequence of real valued random variables with $\mathcal{L}(X_{n})$ (that is, law or distribution) uniform on $[-n,n]$. In what sense(s) do $X_{n}$ converge to a random variable $X$?
We are told that the answer is none. To prove this, are we supposed to show that $X_{n}$ does not converge under the weakest form of convergence possible (convergence in distribution/law)? And if we do, is that sufficient to show that there is no convergence? Also, how does the fact that the r.v.'s are uniformly distributed give us that?

If $U$ is uniform on $[-1,1]$, then $nU$ is uniform on $[-n,n]$. We have for each $t$,
$$\mathbb P(nU\leqslant t)=\mathbb P(U\leqslant t/n)\to 1/2,$$
since the cumulative distribution function of $U$ is continuous at $0$. 
This proves that $(nU)_n$ cannot converge in distribution. 
