So since the l units of stuff in the greedy fractional solution missing from

capital S are a better use of space than the L units of stuff that are in S but

not the greedy fractional solution, some easy algebra shows that indeed the overall

value of the greedy fractional solution is at least the overall value of S.

Since S was arbitrary that shows that the greedy functional solution is in some

sense better than optimal.

It's better than all the non-fractional feasible solutions.

Now it's not clear though why you want to worry bother to do this thought

experiment, which transpired in some fantasy world where we were allowed to

pack items fractionally.

Now what we really care about is our three step greedy algorithm,

and that's not in the fantasy world.

That's in a real world where we cannot pack items fractionally.

So what good could this thought experiment experiment be?

Well, as we'll see in the analysis in the next slide,

it provides a useful yard stick.

A hypothetical bench mark against which we can compare the performance of our

three-step greedy algorithm