1. Let's clarify the concept you mentioned: Cauchy induction is a form of mathematical induction, but it is a bit different from the standard induction. 2. In standard mathematical induction, you prove a base case (often for $n=1$) and then prove that if the statement holds for $n$, it holds for $n+1$. 3. Cauchy induction (or strong induction in some contexts) involves proving the statement for a base case and then assuming it's true for all values less than or equal to some $k$, to prove it for $k+1$. 4. The mention of proving for $2n$ and $n-1$ might be a specific variant or application, but it is not a standard general form of Cauchy induction. 5. If you have a particular theorem or problem context in mind where this approach is used, please share, and I can explain that specific method in detail.