Does the sequence of characteristic functions converge uniformly?

Question

$f_n(x) = 1$ if $x ∈ [0, \frac{1}{n}$] and $f_n(x) = 0$ otherwise. I know that it converges pointwise. My guess is that it doesn't converge uniformly since lim $sup (f_n(x)) = 1$ as n goes to $\infty$. Please correct me if I'm wrong

Answer 1

It converge pointwise to $$f(x)=\begin{cases}1&x=0\\ 0&otherwise\end{cases}.$$

Now, $$|f_n(x)-f(x)|=\chi_{]0,1/n]}(x).$$ Therefore, for all $n$, $$\sup|f_n(x)-f(x)|=1,$$ and thus, the convergence is not uniform.