# Lecture 7

Another common variant is known as *strong induction*. This allows you to assume all previous cases in your induction step, rather than just the last one.

It looks like the following:

**Definition:** [Strong induction] Let $P(n)$ be a statement that depends on a natural number $n$. Then if

$P(0)$ is true, and

for all $k$, if $P(0), P(1), \ldots P(k)$ are all true, then $P(k+1)$ is also true,

then $P(n)$ is true for all $n\in\mathbb{N}$.

Again, this can be viewed as a cleverly disguised version of ordinary induction: if we define the statement $Q(n)$ as follows: $\begin{aligned} Q(n) &= P(0)\wedge P(1)\wedge\cdots\wedge P(n-1)\wedge P(n)\\ &= \text{``all the statements $P(0),\ldots,P(n)$ are true'',}\end{aligned}$ then proving $Q(n)$ by ordinary induction works out to be effectively the same thing as proving $P(n)$ by strong induction.

Indeed, the base case $Q(0)$ is the same thing as the base case $P(0)$. The induction step $Q(k)\Rightarrow Q(k+1)$ looks like $P(0)\wedge\cdots\wedge P(k)\Rightarrow P(0)\wedge\cdots\wedge P(k)\wedge P(k+1).$

In order to prove this, we assume $P(0),\ldots,P(k)$ are all true and have to prove that $P(0),\ldots,P(k+1)$ are all true. But then all of these except the last are assumptions: what is left is to prove $P(k+1)$ assuming $P(0),\ldots,P(k)$, and that’s exactly the induction step of a strong induction.

Here’s an example of strong induction in practice. First we’ll need a definition:

**Definition:** The *Fibonacci numbers* $F_0, F_1,
\ldots$ are defined by taking $F_0 =0$ and $F_1 = 1$, and then for $n\geq 2$ by $F_{n} = F_{n-1} + F_{n-2}.$

Now for the result in question:

#### Proposition

For all $n\in\mathbb{N}$ we have $F_n<2^n$.

#### Proof

Let $P(n)$ be the statement $F_n<2^n$. We know that $P(0)$ is true, since $F_0 = 0<1 = 2^0.$ We also know that $P(1)$ is true, since $F_1 = 1<2 = 2^1.$

Now we’ll show that for all $k\geq 1$ we have that if $P(0),\ldots,P(k)$ are all true, then $P(k+1)$ is true. So suppose that $P(0),\ldots, P(k)$ are all true.

We now have that $\begin{aligned} F_{k+1} &= F_k + F_{k-1}\\ &< 2^k + 2^{k-1}\qquad\text{(using $P(k)$ and $P(k-1)$)}\\ &< 2^k + 2^k\\ &= 2^{k+1},\end{aligned}$ which is exactly $P(k+1)$. This completes the induction step, and so finishes the proof.

That strong induction argument really had *two base cases* before the induction step.

So we proved $P(0)$, and we proved $P(0)\Rightarrow P(1)$ by proving $P(1)$, and then we proved $P(0)\wedge\cdots\wedge P(k)\Rightarrow P(k+1)$ by proving $P(k-1)\wedge P(k)\Rightarrow P(k+1)$.

I like to think that the proof was arranged according to the shape of the definition of the Fibonacci numbers: that definition has two base cases $F_0 = 0$ and $F_1 = 1$, and a step $F_{n+2} = F_{n+1} + F_n$. This is not a rare coincidence.

## Why induction works

In this section we make a few comments on why induction works. They may be helpful in thinking about when you can and when you can’t generalise induction to other settings.

Let’s introduce a definition:

**Definition:** A set (of numbers) is *well-ordered* if every nonempty subset has a least element.

For now, our main use of that is to say this:

**Definition:** The *well-ordering principle for $\mathbb{N}$* says that $\mathbb{N}$ is well-ordered.

This is a very special property of $\mathbb{N}$. The integers $\mathbb{Z}$, for example, are not well-ordered. Indeed, the subset $\mathbb{Z}\subset\mathbb{Z}$ of all integers does not have a least element: there is no least integer.

The main interest is this:

#### Theorem

The well-ordering principle for $\mathbb{N}$ and the principle of strong induction are equivalent.

#### Proof

We’ll show first that we can derive the principle of strong induction from the well-ordering principle.

So suppose we had a statement $P(n)$ for each $n\in\mathbb{N}$, and we had a base case (that $P(0)$ was true) and an induction step (that, for all $k\in\mathbb{N}$ if we have $P(i)$ for all $i<k$, we also have $P(k)$). We need to show that $P(n)$ is true for all $n$.

We might argue as follows. Let $A$ be the set of natural numbers $n$ for which $P(n)$ does not hold: $A = \left\{n\in\mathbb{N}\mid \neg P(n)\right\}.$ So $A$ is the set of “counterexamples”.

If $A$ has any elements at all, it has a smallest element $a$. But $a$ can’t be $0$, because we have $P(0)$. But $a$ can’t be bigger than $0$ either: because $a$ is minimal, we have $P(i)$ for all $i<a$. Hence we have $P(a)$ also, by the induction step. But that’s a contradiction: we assumed that $\neg P(a)$.

Hence $A$ doesn’t have any elements, which is the same as saying that $P(n)$ holds for all $n$.

Now we’ll show the other half of the equivalence: that we can derive the well-ordering principle from the principle of strong induction.

Let $Q(n)$ be the statement, “any subset of $\mathbb{N}$ which contains $n$ has a smallest element”. We’ll prove $Q(n)$ for all $n$ by strong induction.

Firstly, for a base case, we must prove $Q(0)$ (“any subset of $\mathbb{N}$ which contains $0$ has a smallest element”). This is clearly true, as $0$ is the smallest natural number of all, so any such subset has $0$ as its smallest element.

Now we must prove the induction step: we assume for some $k$ that $Q(i)$ is true for all $i<k$, and we prove that $Q(k)$ is true. Consider a subset $S\subset\mathbb{N}$ which contains $k$. If it contains some element $i<k$, then by $Q(i)$ it has a least element. If, however, it contains no element $i<k$, then $k$ is its least element: so in particular, it has a least element. This proves $Q(k)$.

Hence we have $Q(n)$ for all $n$ by strong induction. So if $S$ is a nonempty subset of $\mathbb{N}$, it has at least one element: call it $n$. But then by $Q(n)$, the set $S$ has a least element: this proves the well-ordering principle.

Part of the reason this is such good news is that there are other well-ordered sets, besides the natural numbers. Whenever you find a well-ordering, you get a notion of induction for free.

For example, consider the set of pairs $(m,n)$ of naturals, where we say that $(m,n)<(m',n')$ if $m<m'$, or if $m=m'$ and $n<n'$. (This is called the *lexicographic* ordering, because it’s inspired by the way that words in dictionaries are ordered).

It is not too hard to prove that that is well-ordered: any set of pairs of natural numbers has a “least” element with respect to this ordering. Hence we can do (strong) induction on pairs of naturals!