Abstract Algebra: Theory and Applications · Problem 2.3.4

The induction hypothesis is the assumption you make in the middle of a proof to bridge from one case to the next. Rather than trying to prove your statement for all cases at once, you assume it's true for some arbitrary case $n = k$, then use that assumption to prove it must be true for $n = k+1$. This transforms an impossible-looking infinite task into a single, manageable logical step—the key insight is that once you've shown the base case and this one domino knocks over the next, the whole infinite chain falls.

The induction hypothesis is the assumption you make in the middle of a proof to bridge from one case to the next. Rather than trying to prove your statement for all cases at once, you assume it's true for some arbitrary case $n = k$, then use that assumption to prove it must be true for $n = k+1$. This transforms an impossible-looking infinite task into a single, manageable logical step—the key insight is that once you've shown the base case and this one domino knocks over the next, the whole infinite chain falls.