In this Wikipedia article on the [one dimensional Ising model](https://en.wikipedia.org/wiki/Ising_model#Ising%27s_exact_solution) they use the following factorization

$$
\displaystyle \sum\_{\sigma\_1,\sigma\_2,\cdots,\sigma\_N} e^{a\sigma\_1}
e^{a\sigma\_2}\cdots e^{a\sigma\_N} = 2\displaystyle\prod\_{j=1}^{N}\sum\_{\sigma\_i}e^{a\sigma\_i} = 2 \left[e^{-a}+e^{a}\right]^N \,.
$$
The first sum is over all possible sequences ($\sigma\_1,\sigma\_2,\cdots, \sigma\_N)$ where $\sigma\_i \in \{-1,1\}$.

Does somebody have a reference for this factorization and/or able to give a proof?