  We show that if $Y$ is one of the spaces $\ell^q$, $c_0$, $\ell^\infty$ or
${\textstyle \bigcap_{p > b}} \ell^p$ where $0 < q,b < \infty$, and the
Fr\'{e}chet space $\textstyle \bigcap_{p > a} \ell^p$ is contained in $Y$
properly, then $\textstyle \bigcap_{p > a} \ell^p$ first shows up in the Borel
hierarchy of $Y$ at the multiplicative class of the third level. In particular
$\textstyle \bigcap_{p > a} \ell^p$ is neither an $F_\sigma$ nor a $G_\delta$
subset of $Y$. This answers a question by Nestoridis. This result provides a
natural example of a set in the third level of the Borel hierarchy and with its
help we also give some examples in the fourth level.
