Given the following:
- $\sum_{i=1}^{n} a_i = 1, a_i \geq 0\;\forall i$,
- $\sum_{i=1}^{n} b_i = 1, b_i \geq 0\;\forall i$,
- $\sum_{i=1}^{n} |a_i - b_i| \leq e$ where $e \ll 1$,
what is the upper bound on:
\begin{equation*} \sum_{i,j=1}^{n} |a_i\cdot a_j - b_i\cdot b_j|? \end{equation*}
I am almost sure it is bounded by $2e$ but am not able to prove it.