Non-Existence of the Mean for the Cauchy Distribution

The standard Cauchy distribution has PDF $f(x) = \frac{1}{\pi(1 + x^2)}$ for $x \in (-\infty, \infty)$.

(a) Show that $E[|X|]$ does not exist by proving the integral $\int_0^\infty \frac{x}{\pi(1+x^2)}\,dx$ diverges.

(b) What does this imply about the applicability of the Law of Large Numbers to the sample mean $\bar{X}_n = \frac{1}{n}\sum_{i=1}^n X_i$ when $X_i$ are i.i.d. Cauchy?