= Solution
Again let $A=\{a_1,\ldots,a_m\}$ with $m>1$, and enumerate $B=\{b_1,b_2,\ldots\}$. For each infinite <binary sequence> $x=(x_j)_{j\geq1}$, define a function $g_x$ by
$$
g_x(b_i)=a_i\quad(1\leq i\leq m),\qquad
g_x(b_{m+j})=
\begin{cases}
a_1,&x_j=0,\\
a_2,&x_j=1.
\end{cases}
$$
The first $m$ values make every $g_x$ a <surjective function>, while the remaining values recover $x$ uniquely. Hence $x\mapsto g_x$ is an <injective function> from the uncountable set of <binary sequences> into $S_2$. This proves \b[$S_2$ is an <uncountable set>], an instance of <uncountability of surjections onto a nontrivial finite set>.
Back to article page
