  We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The
first inequality involves cardinal invariants associated with an analytic
P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density
$0$. We obtain an upper bound on the $\ast$-covering number, sometimes also
called the weak covering number, of this ideal by proving in Section
\ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq
\mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship
between the bounding and splitting numbers at regular uncountable cardinals. We
prove in sharp contrast to the case when $\kappa = \omega$, that if $\kappa$ is
any regular uncountable cardinal, then ${\mathfrak{s}}_{\kappa} \leq
{\mathfrak{b}}_{\kappa}$.
