Following is a theorem from Baker Graves Morris book on Padé Approximants :
The proof is not clear. I have two questions :
1- How there are polynomials $p_M(z)$ and $q_M(z)$ of degree at most $M$ such that the equality in the third line of the proof holds?
2- How the last equality holds, i.e. how $...=O(z^{2M+1})$?