Answer:
Hi! So I like your thesis; it sounds good. Honestly, I don't think you need the word "further" in there, unless you've already done an essay or project where you discussed the american dream being improbable.
I think just saying, "The Great Gatsby proves that the American dream is improbable" is much more concise and more powerful.
I think you could also incorporate different cultural aspects during the time period, to prove your point.
Hope this helps!
LMK if you have any other q's!