Give an example of random variables $X_n$ with densities $f_n$ so that $X_n\Rightarrow $ a uniform distribution on $(0,1)$ but $f_n(x)$ does not converge to $1$ for any $x\in [0,1]$.
I tried a lot of examples, but none of are working. I might missing some easy example. Thanks for any help.
I tried $x^{(1+1/n)}$, $x(1-1/n)^x$ etc.
$\Rightarrow$ means weak convergence.