An induction proof has two parts: the base case and the inductive step. Most students understand why the inductive step matters — without it, the chain would not propagate. What gets under-emphasised is the opposite failure: what happens when the inductive step is perfectly fine, but the base case is absent?
The answer is visible in dominoes. You can set up a row of tiles with flawless spacing — every tile, if pushed, will definitely knock the next. But if no one ever pushes the first tile, the chain never starts. The entire line stands upright forever, despite the inductive step being in perfect working order.
The animation below is that exact picture: the inductive step is correct, the dominoes are spaced right, but there is no initial push. You can drag the "push strength" slider all the way up, and nothing happens — because the slider controls a push that was never supposed to be applied. The base case is what applies the push.
The no-push visualisation
Why the inductive step alone proves nothing
The inductive step, in symbols, is the conditional: P(k) \Rightarrow P(k+1). A conditional is only informative when its hypothesis is known to be true. "If it is raining, the streets are wet" tells you something about wet streets provided you also know it is raining. If you never establish the rain, the conditional is sitting idle — consistent with the streets being wet or dry.
An induction proof that skips the base case is exactly the same structure: you have proved a chain of conditionals P(1) \Rightarrow P(2), P(2) \Rightarrow P(3), P(3) \Rightarrow P(4), \dots, but you have not established that P(1) is true to begin with. All those conditionals are sitting idle, waiting for an input that never arrives.
Why this is a real failure mode and not just a pedantic worry: there are genuinely false statements where the inductive step is valid for every k but the base case fails. Consider the claim "for every positive integer n, n = n + 1." The inductive step is trivially valid — if k = k + 1, then adding 1 to both sides gives k + 1 = k + 2, which is P(k+1). So the step goes through. But P(1) says 1 = 2, which is false. Without checking the base case, you could "prove" absurdities. The base case is the gatekeeper that refuses to let false statements ride the inductive chain into apparent truth.
A concrete absurd proof that fails only at the base case
Run induction on the false claim: "every positive integer is larger than 10."
- Inductive step. Assume P(k): k > 10. Then k + 1 > 11 > 10, so P(k+1) holds. The step is valid — in fact, it works for every k.
- Base case. P(1): is 1 > 10? No.
This "proof" has a perfect inductive step and a failing base case. If a student skipped the base case and wrote only the inductive step, they would conclude "by induction, every positive integer is greater than 10," which is obviously false. The base case is what catches the absurdity at the entry point.
This is why competition graders and textbook authors insist on writing the base case out explicitly, even when it seems obvious. The visibility of the base case check is part of the proof's correctness — not just formality.
The base case sets the starting index
A subtler version of "missing base case" is when the base case is present but at the wrong index. If your statement is really only true for n \geq 3, and you check the base case only at n = 3 but write the conclusion as "for all n \geq 1," your proof is corrupt at n = 1 and n = 2.
The rule is simple: the conclusion of the induction is "for all n \geq n_0," where n_0 is the index at which you verified the base case. If you verified P(3), you have proved the statement for n \geq 3, not n \geq 1. Stating it as "for all n \geq 1" without checking P(1) and P(2) separately would leave the first two values unproven.
What the visualisation says about real proofs
The animation compresses an important fact: the inductive step is a transport mechanism, not a source. It moves the truth of P(k) to the truth of P(k+1), but it cannot manufacture truth from nothing. The base case is the source. Without a source, the transport is just empty infrastructure.
When you are reading an induction proof — your own, a classmate's, a textbook's — the base case check is the load-bearing first line. Skip it, misread it, assume it, and the rest of the proof is built on air.
The one-line takeaway
No base case, no induction. The inductive step is the mechanism for propagating truth; the base case is the seed that truth starts from. A perfect mechanism with no seed produces nothing.
Related: Mathematical Induction · Domino Chain Animation — Push One, Watch Induction Reach n = 50 · Missing Inductive Step — The Domino Chain With a Gap at Tile 7 · Proof by Contrapositive · Mathematical Proof — Direct Proof