Question: Hello. I am currently searching for some nice examples of proofs by induction in the finite case, that can be generalized to the infinite case using transfinite induction (and dont become trivial there).

Apparently, the only proof that comes in my mind is that every Vector-Space has a base (mostly this is proven by the Zorn-Lemma, but in the finite case this can be proven by induction, and the same proof can be used in the infinite case with transfinite induction).

Do you have any other examples?

Answer: 
One way to prove the general Van Kampen Theorem is to prove it first for coverings by finitely many open sets, and then generalize to the arbitrarily large coverings using transfinite induction. Whether this is simpler than Hatcher's direct argument for arbitrary coverings (Theorem 1.20 in his Algebraic Topology book) is probably a matter of taste.
Determinacy for $\Delta^0\_1$-games in Baire space is naturally proven by transfinite induction (over the (rank of the) well-founded tree describing the $\Delta^0\_1$-winning set and its complement). This generalizes the backwards-induction argument for finite sequential games having subgame-perfect equilibria.