Since $A$ is finite, there is a bijective function $g:A\rightarrow\{1,\ldots,n\}$ . There is also a bijective correspondence $h$ between the set of all functions $\{f\}$ and $B^{n}$ , namely, $h(f)=(f(g^{-1}(1)),...,f(g^{-1}(n)))$ (basically, we use $g$ to order the elements of $A$ , $a_{i}=g^{-1}(i)$ , and then $h$ lists the images of $a_{1},\ldots,a_{n}$ ). We can now use Corollary 6.8, to argue that, since $B$ is finite, $B^{n}$ is finite, and so is the set of all functions $\{f\}$ (using the composite of $h$ and a bijective correspondence of $B^{n}$ with a section of the positive integer numbers).