Pointwise convergence if $f_n(x)=0$ whenever $x\notin\left(\frac1{n+1},\frac1n\right)$

Given $(a_{n})\in\mathbb{R}$ let$f_{n}:[0,1]\to\mathbb{R}$s.t. $f_{n}=a_{n} $if $\frac{1}{n+1}<x<\frac{1}{n}$and $f_{n}=0$ otherwisefor every $n\in\mathbb{N}$.
I need to find the pointwise convergence of $(f_n)$
I can see how $f_n\to0$ since $\operatorname{diam}f^{-1}(\{a_n\})\to0,n\to\infty$ but don't know whether that observation is usable at all.