Definition of the limit of a sequence I'm looking over the following definition of convergent limits: 
A sequence $(x_n)$ in $\mathbb{R}$ is said to converge to $x \in \mathbb{R}$, or x is said to be a limit of $(x_n)$, if for every $\epsilon > 0$ there exists a $K \in \mathbb{N}$ such that for all $n \geq K$, the terms $x_n$ satisfy $|x_n - x| < \epsilon$.
I'm actually having a lot of trouble understanding how the definition shows that a sequence is convergent. Could someone put it in pretty basic terms?
Also, I'm trying to work through an exercise which states : Prove that $$x_n = (-1)^n + \frac{1}{n}$$ does not converge to 1.
I understand that we need to provided a counter example in which we state that there exists an $\epsilon_0$ such that there exists an "$n$ = something" such that $|x_n - x| \geq \epsilon_0$
But I'm having trouble coming up with an $\epsilon_0$ and $n$ which satisfies the condition $|x_n - x| \geq \epsilon_0$. Could anyone give me a hint?
Thanks

Suppose that $\lim_{x\to \infty}x_n$ exists. Then  $\lim_{x\to \infty}(-1)^n=\lim_{x\to \infty}(x_n-\frac1n)$ exists, which is a obvious contradiction, as we know that, $\lim_{x\to \infty}(-1)^n$ does not exist!
