Product of sums equal to sum of products

Question

Is

$(\sum_k x_k)(\sum_i y_i)(\sum_j z_j)(\sum_l a_l) = \sum_{kijl}x_ky_iz_ja_l$

with

$\sum_{kijl} = \sum_k\sum_i\sum_j\sum_l$?

Answer 1

Let us prove of for two sums.

When you distribute the product over the left sum you get $$ \left( \sum_i x_i\right) \left( \sum_j y_j\right) = \sum_i \left( x_i \sum_j y_j\right) $$ and, when you distribute on the second sum you get $$ = \sum_i \left( \sum_j x_i y_j\right) $$ which is the result (you can drop the $(.)$s as the sum is associative).