Bernoulli Numbers are Rational

Theorem
The Bernoulli numbers are rational.

Proof
By the recurrence relation for the Bernoulli numbers:


 * $B_n = \begin{cases} 1 & : n = 0 \\

\displaystyle - \sum_{k \mathop = 0}^{n-1} \binom n k \frac {B_k} {n - k + 1} & : n > 0 \end{cases}$


 * $B_0$ is rational.


 * $B_n$ is a finite sum of the products of a binomial coefficient with a Bernoulli number earlier in the sequence divided by an integer.

The proof proceeds by induction.

For all $n \in \N$, let $P \left({n}\right)$ be the proposition:
 * $B_n$ is rational.

$P \left({0}\right)$ is true, as this just says:
 * $B_0$ is rational.

Basis for the Induction
$P \left({0}\right)$ is the case:
 * $B_0$ is rational.

This is true, as $B_0 = 1$ which is rational.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $P \left({j}\right)$ is true, where $0 \le j \le k$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is the induction hypothesis:
 * $\forall j \in \N: 0 \le j \le k: B_j$ is rational

from which it is to be shown that:
 * $B_{k + 1}$ is rational.

Induction Step
This is the induction step:

By the recurrence relation for the Bernoulli numbers:


 * $\displaystyle B_{k + 1} = - \sum_{j \mathop = 0}^k \binom {k + 1} j \frac {B_j} {k - j + 2}$

We have that:


 * From Binomial Coefficient is Integer, $\binom {k + 1} j$ is an integer throughout.


 * From the Induction Hypothesis, $\forall j \in \N: 0 \le j \le k: B_j$ is rational.


 * $k - j + 2$ is an integer.

Thus as $B_{k + 1}$ is the sum of a finite number of rational numbers, it follows that $B_{k + 1}$ it itself rational.

The result follows by the Second Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N: B_n$ is rational.