Let $S_n, n = 1,2, \cdots$ denote the partial sums of i.i.d. random variables with positive, finite mean and with a finite moment of order $r, 1 \leqq r < 2$. Let $Z_n, n = 1,2, \cdots$ denote the partial sums of i.i.d. random variables with a finite moment of order $r, 0 < r < 2$, and with mean 0 if $1 \leqq r < 2$. Let $N(c) = \min \{n; S_n > c\}, c \geqq 0$. Theorem 1 states that $N(c)$, (suitably normalized), tends to 0 in $r$-mean as $c \rightarrow \infty$. The first part of that proof follows by applying Theorem 2, which generalizes the known result $E|Z_n|^r = o(n)$, as $n\rightarrow \infty$ to randomly indexed partial sums.