For any sequence of events $E_{1}, E_{2}, \ldots $ (a) If $P\left(E_{n} ight)=0$ for all $n$, show that $P\left(\bigcup_{n=1}^{\infty) E_{n} ight)=0$. (b) If $P\left(E_{n} ight)=1$ for all $n$, show that $P\left(\bigcap_{n=1}^{\infty) E_{n} ight)=1$. (c) Show that $\liminf _{n ightarrow \infty) E_{n} \subseteq \limsup_{n ightarrow \infty) E_{n} $. (d) Show that $P\left(\liminf _{n ightarrow \infty) E_{n} ight) \leq \liminf _{n ightarrow \infty) P\left(E_{n} ight) \leq \limsup_{n ightarrow \infty) P\left(E_{n} ight) \leq P\left(\limsup _{n ightarrow \infty} E_{n} ight) .$ Conlcude that if $\liminf _{n ightarrow \infty) E_{n}=\lim \sup _{n ightarrow \infty) E_{n}=\lim _{n ightarrow \infty) E$, then $P\left(\liminf _{n ightarrow \infty) E_{n} ight)=$ $\lim _{n ightarrow \infty) P\left(E_{n} ight) $. SP.SD.001