The common proof to show that a locally envy-free assignment of slots to bidders (as in a sponsored search auction) is stable is to show that a bidder has a better utility with its current assignment than with the assignment corresponding to him/her being assigned to a better slot (it is easy to show that being assigned a lower slot results in a lower utility).
That proof is often of the form:
By locally envy-freeness, we have:
$v_i*a_i - p_i >= v_i*a_{i-1} - p_{i-1}$
$v_{i-1}*a_{i-1} - p_{i-1} >= v_{i-1}*a_{i-2} - p_{i-2}$
...
$v_{m+1}*a_{m+1} - p_{m+1} >= v_{m+1}*a_m - p_m$
where $v_i$ is the value, $p_i$ the price and $a_i$ the rate/quantity.
Also since a locally envy-free assignment implies an assortative assignment we have:
$v_i < v_{i-1} < v_{i-2} ...$
$a_i < a_{i-1} < a_{i-2} ...$
I understand the above, but the next step I don't. The next step is:
We can then sum up the inequalities and get: $v_i*a_i - p_i >= v_i*a_m - p_m$
What allows me to reach the final statement (we never have $v_i*a_m$ in the original inequalities) and what is the intuition behind it?
Note: The suggested proof can be found here: http://courses.cms.caltech.edu/cs144/lectures/gsp.pdf
