A simple proof of almost percolation on G(n;p)

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.