Verify the following two convergence in distribution results

Verify the following two convergence in distribution results. Suppose that X_n has the discrete uniform distribution on {0, 1/n, .. ., n - 1/n}, n = 1, 2. .. .Show that X_n rightarrow^d U[0, 1], n rightarrow infinity. Let X_n Tilde X^2_n. Show that X_n - n/Squareroot 2n rightarrow^d N(0, 1), n rightarrow infinity.

Solution

Please note that the given problem is related to Probability Distribution theory. Kindly post the question in appropriate medium. However, I have provided the definition of convergence in Distribution below:

Suppose that (X1,X2,…) and X are real-valued random variables with distribution functions (F1,F2,…) and F, respectively. We say that the distribution of Xn converges to the distribution of X as n if Fn(x)F(x) as n for all x at which F is continuous.

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