Using central limit theorem for approximation

#1
I am working on a past exam question and I am not so sure whether or not my answer is correct. If anybody could give me any guidance i would be most appreciated. I have attached the question and my solution in a jpeg file.
 

Dason

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#2
You aren't assuming that X is normally distributed. The point of that second part is to write the product as a sum of the Us and then apply the CLT to that sum.