Newton's iterative method for a scalar equation $f(x)=0$, is known to be convergent to the unique zero $\alpha$ of $f$, for $f$ satisfying certain conditions involving $f$, $f'$ and $f''$. In thi case, if $(x_n)$ is the sequence defined by Newton's method, a stopping inequality or an aposteriori estimation of the form $|x_{n+1}-\alpha|\leq|x_n-x_{n+1}|$, holds. The aim of this paper is to show that Newton's method converges under weaker conditions, involving only $f$ and $f'$, when a generalized stopping inequality is valid.