Does a limit exist for the fraction of $p\alpha_i$?

Question

With $p$ an integer and $\alpha_i$ a real number, does $\{p\alpha_i\}\leq\epsilon$ hold for some predefined $\epsilon$, maybe depending on the number of $\alpha_i$ values? I looked at the simultaneous version of the Dirichlet's approximation theorem (Wikipedia), but the absolute operator is 'in the way'. Or is such a limit impossible?