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 $\limint -In ightarrow \infty) E_in] \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 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 $\limint -In ightarrow \infty) E_in] \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