Examples of problems that are easier in the infinite case than in the finite case. I am looking for examples of problems that are easier in the infinite case than in the finite case. I really can't think of any good ones for now, but I'll be sure to add some when I do.

Stability analysis of finite difference stencils
GKS stability requires that the two semi-infinite problems defined on $[a,\infty]$ and $[-\infty,b]$ must be stable in order for the finite problem defined on $[a,b]$ to be stable. Thus, you can't do the finite case without at least doing the infinite ones first.
