Lemonade Inequality

... reminds me of the kid whose task it is to fill a dozen lemonade glasses on the Thanksgiving dinner table.
(He is also told not to touch any of the glasses.)

When the pitcher is empty, he notices that the last glass has more lemonade than the next-to last one (probably because the pitcher became empty too fast). So, to correct this problem, he gets a straw and drinks up enough lemonade from the more-full glass, to even them out.

Then, he notices that another glass has more lemonade than the last one, so he drinks from that glass, just to even things out. (Perhaps he drinks a bit too much, so he must return to the other two glasses and sip some more from them, to achieve lemonade equality.) Next, he notices that the first glass has even more lemonade than the others, so -- sip, sip, sip -- and now they are equal. Carefully checking each remaining glass, to make sure they have no more than "their share" he sips from each, as necessary. At the other end of the table, he spies a glass that seems to be a bit more full than the one in front of him (possibly due to evaporation!) This recursive process continues until he is satisfied that there is no single glass may contain any more lemonade than any other!

Ultimately, the guests all sit down to precisely-equal glasses that are all nearly empty.