In a recent analysis class at Stanford, we learnt about the neat Riemann series theorem which states:

Theorem 1.1 (Riemann Series Theorem): If an infinite series is conditionally convergent, then you can rearrange the terms in a permutation so that the new series converges to any real number.

It is a fairly interesting result with a really cool proof! However, while messing around with it, I noticed it has a pretty interesting application in constructing injections between certain sets. In particular, it can be use to show that the set, $\mathcal{B}$ of bijections from $\mathbb{N}$ to $\mathbb{N}$ has the same cardinality as $2^\mathbb{N}$.

Theorem 1.2: Let

Then $|\mathcal{B}| = 2^\mathbb{N}$. An interesting (interpretation) of this is that, $\mathbb{N}! = 2^{\mathbb{N}}$ since factorial is often defined in set theoretic terms as the set of bijection from a cardinal to itself.

Proof. We will show $|\mathcal{B}| \leq 2^\mathbb{N}$ and $2^\mathbb{N} \leq |\mathcal{B}|$ and this will give us the result from the Cantor-Schroder-Bernstein Theorem.

The fact that $|\mathcal{B}| \leq 2^\mathbb{N}$ is obvious by the fact that $|\mathcal{B}| \leq \mathbb{N}^\mathbb{N} = 2^\mathbb{N}$.

To show $2^\mathbb{N} \leq |\mathcal{B}|$, we construct an injection from $\mathbb{R} \to \mathcal{B}$ (since $|\mathbb{R}| = 2^\mathbb{N}$). This is where the neat analysis trick comes in. Take a series that converges conditionally e.g.

Then by Riemann’s theorem, for any real number $r$, there is a permutation/rearrangement $\pi : \mathbb{N} \to \mathbb{N}$ of the terms $x_n$ such that $\sum_{n=0}^{\infty} x_{\pi(n)}$ converges to $r$. So given a real $r$, we can injectively map it to the permutation of $\mathbb{N}$ that causes the series to converge to $r$. Since a perumation is equivalent to a bijection, this gives us the desired injection and thus $2^\mathbb{N} = |\mathbb{R}| \leq |\mathcal{B}|$ as required. So by CBS $|\mathcal{B}| = \mathbb{N}! = 2^\mathbb{N}$.