In what way are the main characters of "A & P" and "Araby" initiated into adulthood?
This is my thesis so far:
In the two stories, both authors initiate the two characters into adulthood by their own fears of their very own childhood; nonetheless, marking it the basis of their adulthood.
I think you obviously have a few good ideas to begin with, but your thesis statement you have supplied doesn't appear to me to be as clear and concise as it should be. Let me offer a few ideas for you to think about so that you can perhaps revise it.
One of the central links between these two brilliant short stories is the way that both protagonists experience an epiphany (a sudden insight into their actions, motives and characters) at the end of each story that is linked to their previous actions. Thus it is that the narrator of "A & P," as he leaves his job, reflects:
Looking back in the big windows, over the bags of peat moss and aluminum lawn furniture stacked on the pavement, I could see Lengel in my place in the slot, checking the sheep through. His face was dark grey and his back stiff, as if he's just had an injection of iron, and my stomach kind of fell as I felt how hard the world was going to be to me hereafter.
Having made his "stand," based on his principles and values, which of course, bring him into conflict with the rest of the world, Sammy realises the difficult path he has chosen. To continue being true to himself, he will find that life as a result will not be easy and he will be placed in other areas where he has to make difficult choices. Going against the flow is never easy--nor is trying to recoup your mistakes.
Likewise in "Araby," the unnamed narrator realises just how naive and romantic he has been when he reaches the bazaar and experiences the dull reality of something that his dreams and notions had transformed into an Arabian bazaar full of mystery and magic:
Gazing up into the darkness I saw myself as a creature driven and derided by vanity; and my eyes burned with anguish and anger.
Here, the protagonist realises how stupid he has been and how his illusions and dreams have tainted his understanding of reality. We recognise that from this point on he will have a more pragmatic view of life and not be so quick to be overcome by his dreams.
Thus I would try to revise your thesis statement to incorporate the way that epiphany functions in both texts to initiate the main characters into adulthood. Good luck!