How does one begin to prove the renormalizability of a theory? Power-counting can give you a big clue about renormalizablility, but it's not a proof. In fact, there is only one certain thing that power-counting tells you: coefficients of negative mass dimension are not renormalizeable. But if all the coefficients have positive or zero mass dimension, how does one ensure that all divergences can be canceled by a finite amount of counter-terms of the form of the original Lagrangian? There are so many ways divergences can happen (corresponding to the variety of loop diagrams you can have), yet all these complicated loop integrals have their divergences in a simple form that can be cancelled with a simple counter-term. How does one begin to prove that this is true?