Iterated sums identity

Question

How to show that the following iterated sums are equal?

$\sum_i \sum_j f(i)h(j)g(i,j) = \sum_j\bigg(\sum_i g(i,j)f(i)\bigg) h(j)$

Answer 1

By the definition of the sums? If you open the brackets in the RHS you get the LHS. If that's unclear, you may want to prove this by induction on $j$. For $j = 1,\dots,1$ it is trivial. Suppose that it holds for $j = 1,\dots,n$ and consider adding the $(n+1)$-th term.