1. The **Cauchy induction principle** is a form of mathematical induction useful for proving statements involving sums or sequences that depend on two indices, generally $P(n,m)$. 2. It differs from ordinary induction by proving $P(n,1)$ for all $n$, then assuming $P(n,m)$ holds for some $m$ and proving $P(n,m+1)$ for all $n$. 3. In simpler terms, you start with the base case for $m=1$, then assume the statement is true for a fixed $m$ to show it holds for $m+1$, simultaneously for all $n$. 4. This method is helpful when $n$ and $m$ are intertwined, like proving properties in two-dimensional arrays or sums indexed by $(n,m)$. 5. The steps for a typical Cauchy induction proof: 1. Prove the base case $P(n,1)$ for all $n$. 2. Assume $P(n,m)$ is true for some $m$ and all $n$ (inductive hypothesis). 3. Use this assumption to prove $P(n,m+1)$ for all $n$. 4. Conclude $P(n,m)$ holds for all $n$ and all $m$ by the induction principle. 6. Example: Proving a double summation formula or an identity over a grid of indices with this approach is more natural and concise. In summary, Cauchy induction is a two-dimensional induction method perfect for problems involving two parameters, incrementing one while keeping the other general.