How is the American Dream portrayed in The Great Gatsby?

I feel that the American dream is the constant chase or
strive for that certain success you desire. And when you do complete that ambition is it
just like feeling up a bag with a hole in it "vanity of vanities"??? The unconscious
obsession with greed.

I'm not really sure but it is in the
sense that the Americana dream will not be fulfilled (or the dream is a constant race
towards your own dream not the "American") because of the general American notion for
greed. Meaning when you accomplish a goal, you tend to want more or some new task to
keep yourself busy? Is that why Fitzgerald killed of Gatsby because he wanted to get
across that when he ACCOMPLISHED Daisy he would in the sense get bored with the desired
ambition to win her back. Thus causing him to continue is strive for something that will
satisfy him at that moment or is that death main resultant of the American
dream...?