Show that snnbspalso follows a cauchy distribution- what

Question

Let   be a sequence of IID Cauchy random variables with

(a) Show that s_n also follows a Cauchy distribution.

(b) Prove that in this case, the sample mean does not converge in probability and therefore the weak law of large numbers does not apply. What assumption has been violated in this case that makes the weak law of large numbers not applicable?