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The barrel will never be renewed completely, at least mathematically. The original contents will step wise be diluted to where the original content's fraction is insignificant but not zero. It will approach zero but never equal zero.

An engineer and a mathematician were placed at one end of a dark room, with a beautiful woman at the other end. They are told they can only approach half the distance to the woman every minute. The mathematician storms off grumbling, and as he exits the door he turns to the engineer and asks him if he's coming or not.

The engineer replies "Of course not!"

So the mathematician says "Don't you see you fool, the distance is only halved each time! You'll never reach her!"

With a big grin the engineer says "True, but soon enough I'll be close enough for all practical purposes!"