# a proof on the upperbound of this expression:

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.

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Hence Proved.

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