  Given a nef and big line bundle $L$ on a projective variety $X$ of dimension
$d \geq 2$, we prove that the Seshadri constant of $L$ at a very general point
is larger than $(d+1)^{\frac{1}{d}-1}$. This slightly improves the lower bound
$1/d$ established by Ein, K\"uchle and Lazarsfeld. The proof relies on the
concept of successive minima for line bundles recently introduced by Ambro and
Ito.
