The definition of convergence of a sequence. Suppose $x_1,x_2,\dots\in [-\infty,+\infty]$ and $x\in [-\infty,+\infty]$.

Question

I am reading "An Introduction to Measure Theory" by Terence Tao.

In this book, there is no definition of convergence of a sequence.

Let $[-\infty,+\infty]$ be the extended real number line.
Let $x_1,x_2,\dots\in [-\infty,+\infty]$.
Let $x\in [-\infty,+\infty]$.
What is the definition of $\lim_{n\to\infty}x_n=x$?
If $x_1,x_2,\dots\in (-\infty,+\infty)$, then I know that definition from Calculus.

Suppose $x_i=-\infty$ for some $i\in\mathbb{N}$ or $x_i=+\infty$ for some $i\in\mathbb{N}$. Is the following definition of mine ok?

1. Suppose $x\in (-\infty,+\infty)$. We write $\lim_{n\to\infty} x_n=x$ if:
   For any positive real number $\epsilon$, there exists a natural number $N$ such that if $n\geq N$, then $x_n\in (x-\epsilon,x+\epsilon)$.

2. Suppose $x=-\infty$. We write $\lim_{n\to\infty} x_n=x$ if:
   For any real number $K$, there exists a natural number $N$ such that if $n\geq N$, then $x_n\in[-\infty,K)$.

3. Suppose $x=+\infty$. We write $\lim_{n\to\infty} x_n=x$ if:
   For any real number $K$, there exists a natural number $N$ such that if $n\geq N$, then $x_n\in (K,+\infty]$.