Don't you mean
since by the SLLN.
X doesn't go anywhere, it's free of n.
But I think you wrote this incorrectly.
Hello everbody; i have a question. i really couldnt understand. i think it is easy for people who know theory. i have answer but help to walkthrough. i go on from the middle of question
X ~ N(0,1)
Y ~ chi_square(n)
W_n = 1/n * X/Y
answer says ; use chebyshev inequality
E(Y/n) = 1
Var(Y/n) =2/n < inf for all n
n-> inf. lim V(Y/n) = 0
so it says; W --> N(0,1) in distribution
what i didnt understand is that where did the X go? is it ignored because it is standard normal distribution? or it seems it just reversed denominator.. i really didnt understand why.. thanks for your help.