There are always two kinds of induction in general:
- Assume that the result holds for $n-1$
- Assume the the result holds for integers $<n$
For the former one, we only need to check one base case; for the latter one, we always need to check two base cases.
I’m learning group theory. Induction is a very useful tool that is always employed. We always prove by induction on group order $|G|$ and assume that results hold for groups of order $<|G|$. But I find that we always check only one base, namely $|G|=1$! Maybe I misunderstood something, but I really don’t remember seeing any proof checking more than one base case.
So what is the point? Could you give me some ideas? Any help would be appreciated!