Do the densities of a Uniform[0,1/n] random variable converge pointwise to zero? I'm trying to think of densities that converge pointwise to a function that is not a density. It seems to me that the only way this is possible is if the densities converge to some constant.
Here is something that I thought of but doesn't quite work as far as I can see.
Let $X_n \sim  Unif[0,1/n]$. Thus $X_n$ has density $f_n(x) = nI_{\{0\leq x \leq 1/n\}}$. Does $f_n(x)$ converge to zero pointwise? How do we deal with limits of functions like this? We can't use product rule of limits because n $\rightarrow$ $\infty$ and we can't use l'Hopitals because indicator functions are not differentiable. 
Whether or not this works, what are some other examples of densities that converge pointwise to a function that is not a density?

For example, you can use the normals mean $0$, variance $n^2$. These flatten out nicely as $n\to\infty$. It is easy to see that 
$$\lim_{n\to\infty} \frac{1}{n\sqrt{2\pi}}e^{-x^2/2n^2}=0$$
for all $x$. 
Your proposed example of density function $f_n(x)$ equal to $n$ on $\left(0,\frac{1}{n}\right)$ and $0$ elsewhere also works. For given any positive $x$, if $\frac{1}{n}\lt x$ then $f_n(x)=0$, since then $x$ is outside the interval $\left(0,\frac{1}{n}\right)$. It follows immediately that $\lim_{n\to\infty} f_n(x)=0$. Thus we get pointwise convergence. Note that the convergence is not uniform.
One can play the same game with most standard continuous distributions. For example, one can imitate the normal example to get a family of exponential distribution densities converging pointwise to $0$.
