  We provide a new proof of the sharpness of the phase transition for
nearest-neighbour Bernoulli percolation. More precisely, we show that
  - for $p<p_c$, the probability that the origin is connected by an open path
to distance $n$ decays exponentially fast in $n$.
  - for $p>p_c$, the probability that the origin belongs to an infinite cluster
satisfies the mean-field lower bound $\theta(p)\ge\tfrac{p-p_c}{p(1-p_c)}$.
  This note presents the argument of \cite{DumTas15}, which is valid for
long-range Bernoulli percolation (and for the Ising model) on arbitrary
transitive graphs in the simpler framework of nearest-neighbour Bernoulli
percolation on $\mathbb Z^d$.
