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Can you make an example of a great proof by induction or construction by recursion?

Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen technique :

  • is vital to the argument;
  • sheds new light on the result itself;
  • yields an elegant way to fulfill the task;
  • conveys a powerful and simple view of an intricate matter;
  • is just the only natural way to deal with the problem.

Here induction and recursion are meant in the broadest sense of the words, they can span from induction on natural numbers to well-founded recursion to transfinite induction, and so on...

Elementary examples are especially appreciated, but non-elementary ones are welcome too!

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Structure Theorem for finitely generated abelian groups. – i707107 Mar 30 '12 at 17:02
My favorite induction proof is the joke proof that all natural numbers are interesting. Clearly 0 is interesting. Suppose m is interesting for all $0\leq m\leq n$. If n+1 were not interesting, then it would be the smallest non-interesting number, which is pretty interesting, a contradiction. So n+1 is interesting. – Benjamin Steinberg Apr 2 '12 at 20:32
@Benjamin: Your proof has earned my first laughter of the day! Nice proof! – Lorenzo Lami Apr 3 '12 at 10:04

12 Answers 12

up vote 20 down vote accepted

A Classic: Fix a positive integer $n$. Show that it is possible to tile any $2^n \times 2^n$ grid with exactly one square removed using 'L'-shaped tiles of three squares.

It serves as a wonderful introductory example to proof by induction. Indeed, the proof can almost be represented with two appropriate figures. Yet, for those just learning induction, it is a significant problem where the application of the inductive hypothesis is far from obvious.

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It's a great nontrivial exercise for those learning induction; it's also a nice, entertaining fact for those who already know induction well. Great answer! – Lorenzo Lami Apr 8 '12 at 22:16
There's no reason to insist $n > 0$; works for $n = 0$ as well! – I. J. Kennedy Oct 28 '12 at 16:54

Tsirelson's space (1974) is a good example from Banach space theory. His space is the completion of a $c_{00}$ (all finitely supported scalar sequences) under an inductively defined norm. The base norm is the sup-norm $\|\cdot \|_0$.

For $n \in \mathbb{N}$ the norm $\|x\|_{n+1}$ norm is defined by

$ \|x\|_{n+1}= \sup\{\frac{1}{2} \sum^k_i \|E_ix\|_{n} \} $

where the supremum is taken over all sets $(E_i)_{i=1}^k$, where $E_i$ is a finite interval in $\mathbb{N}$, $\max E_i < \min E_{i+1}$ and $k \leq E_1$ (here $Ex$ denotes the restriction of $x$ to the coordinates of $E$). The Tsirelson norm is $\|x\|_T = \sup_n \|x\|_n$ and satisfies the following implicit equation

$ \|x\|_T= \max ( \|x\|_0 , \sup \frac{1}{2} \sum^k_i \|E_ix\|_T ).$

The space $T$ does not contain a copy of any $\ell_p$ or $c_0$. This solved a major open problem at the time (I should point out that Tsirelson actually defined the dual of $T$ which also has the property).

The, inductive, method he devised for producing this space eventually lead to the solutions of numerous problems in Banach space theory (way to numerous to mention). Moreover, the `necessity' of the inductive construction to produce spaces not containing any $\ell_p$ of $c_0$ is a problem that has been considered by Gowers as a polymath project (unfortunately not much progress here):

Check out Boris Tsirelson's website for more info on his space:

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Definitely a great advanced example! – Lorenzo Lami Apr 8 '12 at 22:21

1)Proof of Euler's formula, V-E+F=2, with induction on F (number of faces).

2)Backward induction proof of generalized AM-GM inequality.

3)Proof of Heine-Borel theorem using Transfinite Topological induction.

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I first saw "topological induction" in one of the early drafts of Thurston's notes, around 1980. A more recent study of the idea is Iraj Kalantari's "Induction over the Continuum", which can be partly viewed at – John Stillwell Mar 31 '12 at 11:09
Sorry, I might be dense, but as far as I understand, an induction that is "not based on an underlying well ordering" is just a bogus mathematical argument. – André Henriques Apr 1 '12 at 14:36
@John I learnt this method of "topological induction" in Dieudonne's "Calcul infinitesimal", where he uses it to prove for example the uniform continuity of continuous functions on closed intervals. I actually used this in class, and students seemed to like it, visualizing the proof as a zipper that you can zip from a to b without being blocked in the middle... – Sylvain Bonnot Apr 1 '12 at 16:28
@Andre: you are not correct, induction can be performed on any well-founded relation, it need not be linear. For example, proof theorists and computer scientists often use induction on well-founded trees. – Andrej Bauer Apr 2 '12 at 6:30
Following up on "topological induction" or "induction over the continuum," there is a recent paper on it by Pete L. Clark at He traces versions of the idea back as far as papers by Khinchin and Perron in the 1920s. – John Stillwell Apr 2 '12 at 7:09

The $n$-level Tower of Hanoi can be solved in $2^n - 1$ moves.

Not only does induction prove this, it actually shows you the solution!

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Goodstein's theorem hasn't yet been mentioned. A straightforward-looking arithmetic theorem with a surprise proof using transfinite induction. Also (the main interesting characteristic of the theorem), there is NO proof from ordinary first-order Peano arithmetic. It's actually equivalent to the formalized $\Sigma^0_1$-soundness (aka 1-consistency) of PA.

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Nice pick, I should have thought about that (and about other statements not provable in PA)! – Lorenzo Lami Apr 3 '12 at 10:09
Goodstein's theorem, which is a $\Pi^0_2$ statement whose existential quantifier is witnessed by a function growing faster than any provably total recursive function of PA, is not equivalent to the consistency of PA, which is merely a $\Pi^0_1$ statement. I'll fix that. – Emil Jeřábek Apr 3 '12 at 10:35

Simultaneous induction as used in combinatorial group theory, for example in the proof of the Adyan-Novikov theorem providing the counterexample to the General Burnside Problem:

Some nice references about the nuts and bolts of this were supplied by Mark Sapir in an answer to one of my questions about this proof:

A synopsis of Adyan’s solution to the general Burnside problem?

Certainly the simultaneous induction technique is an important idea in constructing such monster groups.

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A problem I enjoyed in my undergraduate algorithms course is as follows:

Suppose you have a computing machine with the following architecture. There are $k$ stacks (for some $k$), input can be pushed onto the first stack, output is popped off of the last, and intermediate operations pop from one stack and push to the next in a line. The top of the stack may also be inspected and compared. Given a permutation of $\{1,\ldots,n\}$ in order as input, how many stacks $k$ do you require to sort the permutation? Describe an algorithm that achieves this bound.

One can prove the bound ($\log_2 n$) by induction, and then just state that this gives a natural recursive algorithm. The same technique was useful for a couple of other problems in a similar vein.

I think this certainly fits the bill of an elegant way to fulfill the task (prove a bound and give an achieving algorithm) in a nice class of cases.

The problem is originally from Knuth Vol. 1, and stack sorting is further elaborated on in this survey.

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The "Ercules and the Hydra" problem, as found in "L. Kirby and J. Paris. Accessible independence results for peano arithmetic. London Mathematical Society, 4:285 293, 1982.".

Using transfinite induction, it is possible to show that Hercules will always kill the hydra (with a finite number of blows) regardless of the strategy chosen to chop off hydra's heads. Moreover, this fact is not provable within Peano Arithmetic.

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The original proof of Van der Waerden's theorem on monochromatic arithmetic progressions comes to mind. Well, the more recent ones too by the way.

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Could you please provide a reference? – Lorenzo Lami Mar 31 '12 at 9:36
See the wikipedia article's_theorem – Johan Wästlund Mar 31 '12 at 10:40

Let $P(p)$ = "there is no natural $q$ such that $(p/q)^2=2$". A simple induction argument shows that P holds for all naturals $p$ and hence that $\sqrt 2$ is irrational. All descent arguments are basically induction.

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Among infinite descents, why did you choose this one? – Lorenzo Lami Apr 1 '12 at 20:14
Is this not the canonical example that sets the stage for the others? – Dan Piponi Apr 2 '12 at 1:43

In pro-algebraic geometry you get to see some nice arguments by induction. For example, M. Kim proves that the continuous cohomology $$H^1(G_{\mathbf{Q}_p},\pi_{1,et}^{uni}(X))$$ is representable by induction on the terms in the lower central series of the $\mathbf{Q}_p$ pro-unipotent algebraic group associated to the etale fundamental group of a curve $X$. Not very surprising, but still crucial for the argument.

For a reference, see page 639 in

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