Proving a mathematical statement true for every one of infinitely many cases — every natural number, say — sounds like it should require infinitely many separate checks. Mathematical induction is a proof technique that gets around that entirely, by checking exactly two things, regardless of how large or infinite the set of cases actually is.
Knock over the first domino, then prove dominoes knock each other over
The classic analogy: to prove an entire line of dominoes will fall, you don't need to check each domino individually. You need exactly two facts: the first domino falls (the "base case"), and any domino falling causes the next one to fall too (the "inductive step"). Together, those two facts guarantee every domino in the line falls, no matter how long the line is — because the base case starts the chain, and the inductive step, once proven true in general, guarantees the chain never stops on its own.
Why the inductive step doesn't check case after case — it proves a pattern once
The crucial subtlety, and the part that often confuses students first learning the technique, is that the inductive step isn't proving "case 5 leads to case 6, then case 6 leads to case 7," one pair at a time. It's proving a single, general statement: assuming the claim is true for some arbitrary case k, the "inductive hypothesis," prove it must then also be true for case k+1. Prove that one general implication, and it automatically covers every consecutive pair in the entire infinite sequence at once, because the same argument applies regardless of which specific number k happens to be.
What we're still unsure about
Induction is a completely rigorous proof method within standard mathematics, not a source of ongoing dispute in itself, but it's worth being precise about its actual scope. Proving a statement true for all natural numbers by induction says nothing, on its own, about statements involving real numbers, infinite sets in other contexts, or claims that don't naturally decompose into a "next case" structure at all. A genuinely common student error is trying to apply induction's logic to situations it was never built to cover, which is less a live mathematical controversy than a persistently tricky point in how the technique gets taught and misapplied.
This sits inside Mathematical Induction, one of eight topics in Discrete Mathematics, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.