Question

Simultaneous induction as used in combinatorial group theory, for example in the proof of the Adyan-Novikov theorem providing the counterexample to the General Burnside Problem:

Some nice references about the nuts and bolts of this were supplied by Mark Sapir in an answer to one of my questions about this proof:

http://mathoverflow.net/questions/48184/a-synopsis-of-adyans-solution-to-the-general-burnside-problem

Certainly the simultaneous induction technique is an important idea in constructing such monster groups.