In pro-algebraic geometry you get to see some nice arguments by induction. For example, M. Kim proves that the continuous cohomology $$H^1(G_{\mathbf{Q}_p},\pi_{1,et}^{uni}(X))$$ is representable by induction on the terms in the lower central series of the $\mathbf{Q}_p$ pro-unipotent algebraic group associated to the etale fundamental group of a curve $X$. Not very surprising, but still crucial for the argument.