Examples regarding the Formal $\epsilon - \delta$ definition of Limits for improper Point $\infty$ and $-\infty$

Question

From the Book 'Problems in Calculus in One Variable' by IA MARON.

Chapter - 1, section 1.6:

1.6 Number Sequences. Limit of Sequence:

The number $a$ is called limit of a sequence $x_1, x_2, ....x_n,...$ as $n \rightarrow 0$, $a = \lim \limits_{n \rightarrow \infty} x_n$ if for $\epsilon>0$, there exists a number $N(e)>0$ such that the inequalitiy $| x_n - a|< \epsilon$ holds true for all $n>N(\epsilon)$

What are $\epsilon$, $N(e)$ and $N(\epsilon)$ here? Can this above theory be explained to me by giving an example, please?

The sense of the above extract from the book is unclear to me. Apologies if this really makes sense if I'm unable to get it.

There are many questions similar around - What is the formal epsilon-delta definition and how do you use to prove a limit

They don't define this for any improper point and My question is specific to this book.

Answer 1

Let $x_{(\cdot)}$ be a sequence of reals. Fix $a\in\mathbb{R}$.

In words: $a$ is the limit of $x_{(\cdot)}$ if for each $\varepsilon>0$, eventually for each $n\in\mathbb{N}$, the distance between $x_n$ and $a$ is less than $\varepsilon$.

In pure symbols: $$\underbrace{(\forall \varepsilon>0)}_{\text{for each $\varepsilon>0$}}\,\,\,\,\underbrace{(\exists N\in\mathbb{N})(\forall n\ge N)}_{\text{eventually for each $n\in\mathbb{N}$}}\underbrace{(|x_n-a|<\varepsilon)}_{\text{the distance is less than $\varepsilon$}}$$

Example: I claim that $\lim_{n\to\infty}\frac{1}{n}=0$. Proof. Fix an arbitrary $\varepsilon>0$. There exists a natural number $N$ such that $1<\varepsilon \cdot N$ (Archimedes Principle). For each $n\ge N$, we have $$|x_n-a|=\frac{1}{n}\le \frac{1}{N}<\varepsilon$$ as desired. That is to say we have shown that for each $\varepsilon>0$, eventually for each $n\in\mathbb{N}$, the distance between $x_n$ and $a$ is less than $\varepsilon$.