Question:Prove that this sequence does not converge for any $x \in \Bbb R.$
$x_n=(-1)^n(1-\frac{1}{n})$
Definition (Negation):$ \exists \varepsilon \forall N \in \Bbb N \exists n \in \Bbb N, n>N: |x_n-x|\geq \varepsilon$
I would usually tackle this question by taking the odd/even sub-sequence and showing that the limits are unequal, however I am struggling with proving it using the negation for the definition of convergence. Mainly, what is the logic of picking epsilon/n?