Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done.
The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n \simeq S_n$, where $$S_n = X_1 + \dots = X_n = \sum_{j=1}^n X_j.$$ Unfortunately, this isn't the case: if it were, then we'd have "$\to Z \sim N(0,1)$", not "$\to 1$". (?) Also, I have, of course, noted that if we define $X^{(n)} = (X_1,...,X_n)$, then we have the scenario of (i), and $R_n = \Vert X^{(n)} \Vert$; whether this turns out to be of help is another matter!
Any advice for part (ii) only would be most appreciated. I would like to think that determining how to do (ii) would then make me able to do (iii) - I can live in hope. :)
Best regards, Sam
PS: This is the last question on the last example sheet for a pretty tough course (for 3rd year undergrad), so I expect the answer won't be massively simple. Alternatively, it could be really simple if you can get it. =D
Hint for (ii): Show that $R_n^2/n\to1$ almost surely; no approximations or additional constructions necessary.