I need a debateable thesis about how growing old and deterioration is apparent in "A Rose For Emily."
You might want to think about how, throughout this excellent and rather chilling short story, Miss Emily is presented as something of an anachronism in her society. It is clear that she is shown to be some form of remnant of an earlier, now obsolete age, and there is a great quote in the first section of this short story that you can use to suggest this idea:
It was a big, squarish frame house that had once been white, decorated with cupolas and spires and scrolled balconies in the heavily lightsome style of the seventies, set on what had once been our most select street. But garages and cotton gins had encroached and obliterated even the august names of that neighbourhood; only Miss Emily's house was left, lifting its stubborn and coquettish decay above the cotton wagons and the gasoline pumps--an eyesore among eyesores. And now Miss Emily had gone to join the representatives of those august names where they lay in the cedar-bemused cemetery among the ranked and anonymous graves of Union and Confederate solders who fell at the battle of Jefferson.
Note how this quote links the character of Miss Emily with her house. Both are relics of an age that has long been passed by and overtaken by more modern equivalents. The decay of the house can find its parallel with the increasing old age and decrepitude in the appearance of Miss Emily as she is viewed later on in the story. Lastly, the vision of Miss Emily joining the graves of those soldiers in the Civil War seem to consolidate the story's focus on death and decay.
Of course, this is just one instance in the story. You need to read through the tale again now to identify other quotes that support this idea. Good luck!