Question: let $X_{n}=\frac{(n-1)(-1)^{n}}{n}$ find the $\limsup(X_{n})$ and $\liminf(X_{n})$
Can I get someone to help me with this proof? I get that $\frac{(-1)^{n}}{n}$ gives me $\liminf(X_{n})=\limsup(X_{n})=0$, but how does it get affected by multiplying it by $(n-1)$? How can I go from here to obtain a proof that flows?
Actually the $\lim \inf$ is $-1$ and the $\lim \sup $ is $1$. To prove it, remark that $-1 \leq X_n \leq 1$ and find a subsequence $n_k$ such that $X_{n_k}$ tends to $1$. Then, do the same for $-1$.