= Solution
Let $M_n=\max_{1\leq j\leq n}W_j$ for <independent and identically distributed random variables> with <distribution function> $F$. Then $P(M_n\leq u)=F(u)^n$. Membership in the <maximum domain of attraction> of a nondegenerate <distribution function> $G$ means that some deterministic $a_n>0$ and $b_n\in\mathbb R$ satisfy
$$
\boxed{\frac{M_n-b_n}{a_n}\xrightarrow{d}Z,\qquad P(Z\leq x)=G(x),}
$$
or equivalently $F(a_nx+b_n)^n\to G(x)$ at every continuity point of $G$.
Under the stated convergence, put $Z_n=(M_n-b_n)/a_n$, $r_n=\alpha_n/a_n\to a>0$, and $s_n=(\beta_n-b_n)/a_n\to b$. The <Slutsky theorem> says that combining a sequence with <convergence in distribution> with sequences converging in <probability> to constants preserves the corresponding sum, product and ratio limits, provided a limiting denominator is nonzero. Applying it here gives
$$
\frac{M_n-\beta_n}{\alpha_n}=\frac{Z_n-s_n}{r_n}\xrightarrow{d}\frac{Z-b}{a}.
$$
The limiting <distribution function> is $P((Z-b)/a\leq x)=G(ax+b)$, continuous for every real $x$ because $G$ is continuous and $a>0$. Consequently,
$$
\boxed{F(\alpha_nx+\beta_n)^n\longrightarrow G(ax+b)\quad\text{for every real }x.}
$$
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