Let $\{a_n\}_{n=1}^\infty$ be a sequence of real numbers in $[0,1]$ and let $\{b_n\}_{n=1}^\infty$ be a sequence of positive real numbers such that $\sum_{n=1}^\infty b_n<\infty$. Show that there exists a number $x\in[0,1]$ such that:
$$\sum_{n=1}^\infty\frac{b_n^2}{|x-a_n|}<\infty$$
My attempt was to let $X\sim U[0,1]$ be a random variable and define events which will allow me to use Borel-Cantelli lemma. However, those are only ideas and I wasn't able to make a lot of progress.
I'll appreciate any help!
Let $A_n:= (a_n-b_n,a_n+b_n)\cap [0,1]$. By assumption, $\sum_{n\geqslant 1}\lambda(A_n)<\infty$ hence by the Borel-Cantelli lemma, almost every $x$ belongs to at most finitely many $A_n$. Moreover, if $x\notin A_n$, then $$ \frac{b_n^2}{|x-a_n|}\leqslant b_n.$$