How can I prove $\sum_{\{n_i\}} \prod_i a_i^{n_i} = \prod_i \sum_{n_i=0}^\infty a_i^{n_i}$? The equality 
$$
\sum_{\{n_i\}}  \prod_i  a_i^{n_i} = \prod_i \sum_{n_i=0}^\infty a_i^{n_i},
$$
is used in some notes on Quantum Field Theory by De Wit. Here the summation on the left hand side is over all possible sequences of non-negative integers and we can take, without loss of generality $i=1,2,3,\ldots$. 
If I write the lowest order terms for the sequences with $a_i=0$ for $i>2$ only, I get, for the left hand side,
$$
1+a_1+a_2+a_1^2 + a_2^2 + a_1 a_2 + a_1^3 + a_2^3 + a_1^2 a_2 +  a_1 a_2^2 + \ldots,
$$ 
and for the right hand side,
$$
 (1+a_1+a_1^2+a_1^3 + \ldots) (1+a_2+a_2^2+a_2^3 + \ldots) ,
$$
which is clearly the same. Therefore I can convince myself that the above equality holds, but I would like to see a real proof. Is there anyone who could give me such a proof? 
If someone knows a better way to phrase the title, such that it is also findable for other users, please let me know.
 A: We have 
$$\prod_{i=0}^\infty\sum_{n_i=0}^\infty a_i^{n_i} = (a_0^0 + a_0^1 + a_0^2 + \ldots)(a_1^0 + a_1^1 + a_1^2 + \ldots)\ldots$$
When multiplying out the brackets on the right we will get a sum of terms $a_0^{i_0}a_1^{i_1}a_2^{i_2}\cdots$ for all possible combinations of $\{i_1,i_2,i_3,\ldots\}$ where $i_k$ are integers so
$$\prod_{i=0}^\infty\sum_{n_i=0}^\infty a_i^{n_i} = \sum_{\text{All possible integer sequences }\{i_1,i_2,\ldots\}} a_0^{i_1}a_1^{i_2}a_3^{i_3}\cdots$$
The above sum is exactly what the author calls $\sum_{\{n_i\}}$ (I choose to use $i_k$ to seperate it from the left hand side summation index). The summand can be written more compactly as
$$a_0^{i_0}a_1^{i_1}a_2^{i_2}\cdots = \prod_{k=0}^\infty a_k^{i_k}$$
and we arrive at the identity
$$\prod_{i=0}^\infty\sum_{n_i=0}^\infty a_i^{n_i} = \sum_{\text{All possible integer sequences }\{i_1,i_2,\ldots\}}\prod_{k=0}^\infty a_k^{i_k}$$
Yes, it is as simple as this, we just need to observe what happens when the product-sum is multiplied out. I might add that I have assumed that all the sequences converge absolutely so that there are no convergence issues here associated with rearrangment.
