Let $\Lambda$ be an artin algebra. We say $\Lambda$ is of finite representation type if the number of the isomorphism classes of indecomposable $\Lambda$-modules is finite.
Now suppose $\Lambda$ is not of finite type, how to get an simple module $S$ such that there is an infinite number of nonisomorphic indecomposable modules $X$ with $Hom_{\Lambda}(X,S) \not =0$?
Since it is an artin algebra there are finitely many simple modules $S_1$,...,$S_n$ up to isomorphism (namely the summands of the module A/J, when J is the jacobson radical). Let $M=S_1 \oplus ... \oplus S_n$. Then for every indecomposable module $N: Hom(N,M) \neq 0$. Now $ Hom(N,M)=Hom(N,S_1) \oplus ... \oplus Hom(N,S_n)$ is always nonzero which would not be possible if for every simple module $S$ Hom(N,S)=0 for all but finitely many indecomposable modules $N$.