Limit in probability of (x-bar)^2 ?

#3
I suppose that is the idea right that the variation between the sample statistics and the population parameter is 0. I'm just not sure how to show what the actual limit in probability is of (x-bar)^2 and if the information about the CDF is relevant to finding the limit in probability
 
#4
It may be more complex than this, but I'm not sure why you can't use the CDF to find the mean of the distribution and - armed with that - simply use the CMT to determine what the the sample mean squared converges to.