In this exercise, we construct an example of a sequence of random variables Zn such that Zn converges to 0 with probability 1, but Zn fails to converge to 0 in quadratic mean. Let X be a random variable having the uniform distribution on the interval [0, 1]. Define the sequence Zn by Zn = n2 if 0 < X < 1/n and Zn = 0 otherwise.
a. Prove that Zn converges to 0 with probability 1.
b. Prove that Zn does not converge to 0 in quadratic mean.