How does The Great Gatsby illustrate the failure of the American Dream?

In The Great Gatsby, Fitzgerald illustrates the failure of the American Dream in two ways. First of all, he does this through the settings he creates. Think, for example, about the differences between West Egg and the Valley of Ashes. West Egg is home to Jay Gatsby's mansion, described as a "colossal affair" which resembles the luxurious Hotel de Ville in Normandy. Despite being "less fashionable" than the neighboring East Egg, West Egg remains one of the most glamorous and wealthiest areas in New York. In contrast, the Valley of the Ashes, first introduced in Chapter Two, is a wasteland dominated by "chimneys and smoke" and "ash-grey men." Here, people like George Wilson live a far less glamorous existence than the likes of Jay Gatsby. Wilson's garage, for instance, reflects his life: it is "unprosperous" and "bare" and is filled with the "dust-covered wreck" of a car. Through this setting, then, Fitzgerald argues that the American Dream has left many people behind. These people, like George Wilson, are left to fester in the Valley of Ashes, forgotten by the rest of society.
Fitzgerald also demonstrates the failure of the American Dream through the character of Jay Gatsby. For Gatsby, the American Dream is closely linked to his feelings for Daisy Buchanan: he is madly in love with her and will do anything to win her back. But Daisy married Tom Buchanan, an extremely wealthy man, with whom Gatsby must compete if he is to convince her of his worth. Over the course of his adult life, Gatsby devoted all of his energy to making as much money as possible and, in this respect, Gatsby represents the success of the American Dream.
Despite his immense material wealth, however, Gatsby fails to win Daisy back permanently. The pair enjoy a brief relationship but Daisy refuses to divorce Tom and, all of a sudden, Gatsby's dream is over. His wealth and his success are therefore inconsequential and the American Dream is portrayed as a failure. To further highlight this idea, when Gatsby dies, his funeral is attended only by his father, Owl Eyes, and Nick, which provides a sharp contrast to his lavish parties. These were the focal point of New York's social life and were attended by scores of people. Through this sad finale, Fitzgerald shows that the American Dream brings only misery and isolation instead of the happiness and success that it first promises.