  We find uniform with respect to parameter $p \ (1\leq p\leq\infty)$ upper
estimations of best approximations by trigonometric polynomials of classes
$C^{\psi}_{\beta,p}$ of periodic functions generated by sequences $\psi(k)$,
that decrease to nought faster than any power function. Obtained estimations
are exact for order and have constants that are written in explicit form and
depend on function $\psi$ only. We obtain analogical estimations for best
approximations of classes $L^{\psi}_{\beta,1}$ in metrics of spaces $L_{s}$,
$1\leq s\leq \infty$.
