Let \(\left\{U_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent \(\operatorname{UNIFORm}(0,1)\) random variables. For each definition of \(A_{n}\) given below, calculate

\[P\left(\limsup _{n ightarrow \infty} A_{n}ight)\]

a. \(A_{n}=\left\{U_{n}

b. \(A_{n}=\left\{U_{n}

c. \(A_{n}=\left\{U_{n}<\exp (-n)ight\}\) for all \(n \in \mathbb{N}\).

d. \(A_{n}=\left\{U_{n}<2^{-n}ight\}\) for all \(n \in \mathbb{N}\).