# Proving Riemann's Theorem through Casorati- Weierstrass

The title is pretty much self-explanatory; I'd like to know if there is a proof of Riemann's rearrangement theorem that uses the Casorati-Weierstrass theorem. Even if you're not familiar with such a proof but can come up with one, I'd like to have a look. To be specific:

Riemann's Rearrangement Theorem

Let $(a_n)\subset\mathbb{R}$ be a real valued sequence such that the series $\displaystyle{\sum_{n=1}^{\infty}a_n}$ converges, but not absolutely, meaning that $\displaystyle{\sum_{n=1}^{\infty}|a_n|=+\infty}$. Then, for every $c \in \overline{\mathbb{R}}$ there exists a rearrangement $k:\mathbb{N}\to\mathbb{N}$ such that $\displaystyle{\sum_{n=1}^{\infty}a_{k(n)} = c}$.

Comment: A rearrangement is just a bijection of the positive integers.

Casorati-Weierstrass Theorem

Let $z_0 \in \Omega$, with $\Omega$ being an open set and $f:\Omega\setminus\{z_0\}\to\mathbb{C}$ be a holomorphic function, such that $z_0$ is an essential singularity of $f$. Then, for each $\delta>0$ that satisfies $D(z_0,\delta)\subset\Omega$, the set$f(D(z_0,\delta)\setminus\{z_0\})$ is dense in $\mathbb{C}$.

I'm familiar with the proofs of both theorems, but I can't seem to make a connection. I'm not even sure there is one; but both of the theorems flirt with the density of the limit points and I strongly believe that they're connected.

• Why would you think there is a connection? – mrf May 30 '17 at 14:13
• Instincts, I suppose – JustDroppedIn May 30 '17 at 14:24