Professor Gekko and his North Dakota adventure:

Professor Gekko's strategy is to choose the most distant place to stop for water within m miles.  (Assuming there is always a water stop within m miles.)

So, let's say that there is a better way to do it (the Poe way), and that better way disagrees with Gekko's first decision.

So, the Poe way has to stop for water before Gekko does.  He can't stop after Gekko, since Gekko has chosen the last possible place to stop.

Now, if Poe changes his first to match Gekko's choice, is it true that Poe has improved his algorithm or at least kept it the same.  And the answer is yes.  Because if Poe changes his first stop to match Gekko's stop, he now has more water at Gekko's stop than he would have had under his previous strategy.

So, if Poe had an intermediate stop between his first stop and Gekko's first stop, he no longer has to make it, so that's an improvement.  Once he makes it to Gekko's first stop than he can totally visit all the other stops in his original algorithm because he has more water now having fueled up at Gekko's first stop.  So, changing Poe's first step to match Gekko's in no way harms Poe's algorithm and might actually improve it.

Let's say Poe's and Gekko's algorithms agree on the first k stops, but disagree on the k+1st.  Well, Poe can do the same thing, he can refuel at Gekko's k+1st stop, which is further down that Poe's, since Gekko fuels at the possible stop.  In doing so, he might skip over some intermediate stops, improving Poe's algorithm, but in any event, after changing his k+1st stop to match Gekko's, Poe can still make all of his water stops because he has more water, having refueled a little later.

Thus, if Poe changes all of his decisions to match Gekko's, then he hasn't harmed his algorithm, and might have improved it (if he gets to skip some stops along the way).  So, Gekko's algorithm has to be the best.