  Let GCH hold and let $j:V\longrightarrow M$ be a definable elementary
embedding such that $crit(j)=\kappa$, $^{\kappa}M\subseteq M$ and
$\kappa^{++}=\kappa_{M}^{++}$. H. Woodin proved that there is a cofinality
preserving generic extension in which $\kappa$ is measurable and GCH fails at
it. This is done by using an Easton support iteration of Cohen forcings for
blowing the power of every inaccessible $\alpha\leq\kappa$ to $\alpha^{++}$,
and then adding another forcing on top of that. We show that it is enough to
use the iterated forcing, and that the latter forcing is not needed. We will
show this not only for the case where $\kappa^{++}=\kappa_{M}^{++}$, but for
every successor ordinal $\gamma$, where $0< \gamma< \kappa$, we will show it
when the assumption is $\kappa^{+\gamma}=\kappa_{M}^{+\gamma}$.
