Given an arbitrary sequence of random variables (or say measurable functions on a finite-measure space) $\xi_n$, one can show by a truncation and Borel-Cantelli argument that there always exists a sequence $c_n>0$ such that $$ \sum_{n=1}^\infty c_n \xi_n \quad \text{converges almost surely.} $$

Can one give an example to show that the "almost surely" in the statement CANNOT be strengthened to "pointwise"?