  We consider bootstrap percolation on the binomial random graph $G(n,p)$ with
infection threshold $r\in \mathbb{N}$, an infection process which starts from a
set of initially infected vertices and in each step every vertex with at least
$r$ infected neighbours becomes infected. We improve the results of Janson, \L
uczak, Turova, and Valier (2012) by strengthening the probability bounds on the
number of infected vertices at the end of the process, using simple arguments
based on martingales and giant components.
