Consider the case where \(\left\{A_{n}ight\}_{n=1}^{\infty}\) is a sequence of independent events that all have the same probability \(p \in(0,1)\). Prove that

\[P\left(\limsup _{n ightarrow \infty} A_{n}ight)=1,\]

and interpret this result in terms of how often the event occurs.