The property of extendability of limit distributions for the maximum term of a sequence

Let $\xi_1,\xi_2,\dots$ be a sequence of identically distributed independent random variables, and let
$$ \eta_n=\max(\xi,\xi_2,\dots,\xi_n). $$
The following theorem is proved: If for a certain choice of constants $b_n>0$ and $a_n$
$$ P\biggl\{\frac1{b_n}(\eta_n-a_n)<x\biggr\}\to\Phi(x),\quad n\to\infty, $$
where $\Phi(x)$ is one of the three possible limiting distributions, and if the convergence is fulfilled in an interval $(c,d)$ for which $\Phi(d)-\Phi(c)>0$, then the convergence holds for all values of $x$.
Библиогр. 5.