The two dimensional Hubbard Model at half-filling: I. Convergent Contributions

We prove analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index $i$ is proved to be analytic in $\lambda$ for $|\lambda| \le c/i$ for some constant $c$, and the skeleton part of the model at temperature $T$ (the sum of all graphs without two point insertions) is proved to be analytic in $\lambda$ for $|\lambda| \le c/|\log T|^{2}$. These theorems are necessary steps towards proving that the Hubbard model at half-filling is {\it not} a Fermi liquid (in the mathematically precise sense of Salmhofer).