I want to prove a hypothesis (H) for all $(m,n)$ where $m,n\in\mathbb{Z}_{\ge 0}$ by induction. I know H holds for base cases $(h,k)\in B := 0\times \mathbb{R} \cup \mathbb{R} \times 0$.
In the induction step, could I assume H holds for $$ (h,k)\in S:= B\; \cup ( \{1,...,m\}\times \{1,...,n\}\setminus (m,n) ) $$ for $m,n\in\mathbb{Z}_{\ge 1}$ and then show that H also holds for $(m,n)?$ In particular here we assumed H holds for both $(m-1,n),(m,n-1)$ but we also know both $(0,1),(1,0)$ are base cases so it seems it may work.
Is this rigorous? or does it depend on the specific of the proof?