Let $\{x_n\}$ be a sequence. If $x_{2n-1}=-\frac{1}{n}$ and $x_{2n}=1+\frac{1}{n}$ and $X_n = \{ x_n, x_{n+1},\ldots\}$, verify that $\inf X_{2n-1} = -\frac{1}{n}$ and $\sup X_{2n-1} - \sup X_{2n} = 1 + \frac{1}{n}$ and prove that $\liminf x_n = 0$ and $\limsup x_n = 1$. Besides, prove that those are the only adherence values of the sequence $\{x_n\}$.
I was able to prove that $\liminf x_n = 0$ and $\limsup x_n = 1$, but I don't know how to start on verifying the first part and proving that they are the only adherence values of the sequence.