Consider a sequence of random variables \(\left\{X_{n}ight\}_{n=1}^{\infty}\) where \(X_{n}\) has probability distribution function

\[f_{n}(x)= \begin{cases}{[\log (n+1)]^{-1}} & x=n \\ 1-[\log (n+1)]^{-1} & x=0 \\ 0 & \text { elsewhere }\end{cases}\]

for all \(n \in \mathbb{N}\).

a. Prove that \(X_{n} \xrightarrow{p} 0\) as \(n ightarrow \infty\).

b. Let \(r>0\). Determine whether \(X_{n} \xrightarrow{r} 0\) as \(n ightarrow \infty\).