Is there a simple proof for this?

Edit: Never mind. I misread the question. :)

I posted a proof here. http://mathbin.net/90747

Thanks,@BonusQuestion. I eventually came up with a similar proof. I also came up with an intuitive way of looking at it. For x < median, most of the values of |Xi – x| will be Xi – x and not x – Xi. Therefore, as x moves from Xi to X(i+1), the net negative coefficient of x will cause f(x) to decrease. The opposite holds true for x > median. The other thing to realize is that the function is continuous, because Xi – x = x – Xi = 0 for x= Xi.

Here’s my way of visualizing the solution:

Imagine a number line with the n points of Xi plotted on it. The goal of the problem is to find a point of the number line where the sum of the distances to the points in the set is minimal. If you pick a value of x such that 7 points are on the right and 3 points are on the left, then you know that x must be moved to the right (increased). If you were to move the point one unit to the right (assuming you don’t cross over any point in the set) then you would be one unit closer to 7 points and one unit farther away from 3 points. This would make the summed distance decrease by 4 units, so you are headed in the right direction. Hopefully this shows how the best direction to move the point on the number line is towards the side with more points. This would balance out when the number on both sides is equal, which is the median of the set.