I cannot figure out how to prove that the sequence $x_n\:=\:\left(-1\right)^n\:+\:\frac{1}{n}$ does not converge. I tried a proof by contradiction but the sequence ends up being $\frac{1}{n}+1\:-\:\frac{1}{n}$ and I don't know where to go from there. I also tried showing that its subsequences don't converge but I did not know what to do with them
$x_{2n} = (-1)^2n + \frac{1}{2n}$
$x_{2n-1} = (-1)^2n-1 + \frac{1}{2n}-1$
Hint: \begin{align*} (-1)^{2n} + \frac{1}{2n} &= 1 + \frac{1}{2n} \\ (-1)^{2n+1} + \frac{1}{2n} &= -1 + \frac{1}{2n} \end{align*} For large enough $n$, one of these is greater than $1/2$ and the other is less than $-1/2$...