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"?



Taking away the "almost sure"
Let $\Omega=(0,1)$ and $\xi\_n:\omega\in\Omega\mapsto$ the $n$-th term of the continued fraction expansion of $\omega$. Given a sequence $c\_n$, there is another sequence $m\_n\in\mathbb{N}$ such that $\sum\_{n=1}^\infty c\_nm\_n$ diverges. Let $x=[m\_1,m\_2,\dots,m\_n,\dots]$. Then $\sum\_{n=1}^\infty c\_n\xi\_n(x)$ diverges.


