Let \(\left\{F_{n}ight\}_{n=1}^{\infty}\) be a sequence of distribution functions such that \(F_{n} \leadsto F\) as \(n ightarrow \infty\) for some distribution function \(F\). Let \(\left\{t_{n}ight\}_{n=1}^{\infty}\) be a sequence of real numbers.

a. Prove that if \(t_{n} ightarrow \infty\) as \(n ightarrow \infty\) then

\[\lim _{n ightarrow \infty} F_{n}\left(t_{n}ight)=1\]

b. Prove that if \(t_{n} ightarrow-\infty\) as \(n ightarrow \infty\) then

\[\lim _{n ightarrow \infty} F_{n}\left(t_{n}ight)=0\]

c. Prove that if \(t_{n} ightarrow t\) where \(t \in C(F)\) as \(n ightarrow \infty\) then

\[\lim _{n ightarrow \infty} F_{n}\left(t_{n}ight)=F(t)\]