Uniformly converging

Question

Can someone help me how to conclude if the following converges uniformly:

$\frac{1}{nx}$$Χ_{(1/n,1]}$$(x)$

where $Χ$ is the characteristic function?

Answer 1

It's not hard to see that $f_n\to 0$ pointwise everywhere. If $f_n$ converged uniformly to $0$ on $\mathbb R,$ we would have

$$\tag 1 \sup_{\mathbb R}|f_n|\to 0.$$

But for each $n>1,$ $f_n(2/n) = 1/2.$ Thus the supremums in $(1)$ are all at least $1/2$ for $n>1,$ showing $(1)$ fails. Thus $f_n$ fails to converge uniformly to $0$