  We show that Mahler's classification of real numbers $\zeta$ with respect to
the growth of the sequence $(w_{n}(\zeta))_{n\geq 1}$ is equivalently induced
by certain natural assumptions on the decay of the sequence
$(\lambda_{n}(\zeta))_{n\geq 1}$ concerning simultaneous rational
approximation. Thereby we obtain a much clearer picture on simultaneous
approximation to successive powers of a real number in general. Another variant
of the Mahler classification concerning uniform approximation by algebraic
numbers is shown as well. Our method has several applications to classic
exponents of Diophantine approximation and metric theory. We deduce estimates
on the Hausdorff dimension of well-approximable vectors on the Veronese curve
and refine the best known upper bound for the exponent
$\widehat{\lambda}_{n}(\zeta)$ for even $n\geq 4$.
