  A short proof of the classic Hardy inequality is presented for $p$-norms with
$p>1$. Along the lines of this proof a sharpened version is proved of a recent
generalization of Hardy's inequality in the terminology of probability theory.
A probability theoretic version of Copson's inequality is discussed as well.
Also for $0<p<1$ probability theoretic generalizations of the Hardy and the
Copson inequality are proved.
