Suppose I want to prove $p(m,n)=1/2$ for all $m\ge1,n\ge1$, and I know the base case $p(1,1)=1/2$ holds.
Here is my induction proof:
- assume $p(m-1,n),p(m,n-1),p(m-1,n-1),p(m-2,n-1)\cdots$ all equals to $1/2$.
- with above assumption, show that $p(m,n)=1/2$.
Is above induction logically rigorous? why and why not?
This question is my analogy to a single variable problem, which is like this:
Suppose we want to prove $p(n)=1/2$ holds for any positive integer $n$, it is easy to show that $p(1)=1/2$. Assuming $p(k)=1/2$ holds for all $k<n$, using this, prove $p(n)=1/2$.
This is the standard way to make an induction proof.