Is there a natural way to define $+\infty$ and $-\infty$ as sets? The extended reals are taken to be the set $\mathbb{R}\cup\{+\infty,-\infty\}$. But is there a natural way to define $+\infty$ and $-\infty$ as sets, in a pure set-theoretic theme, following the construction of $\mathbb{R}$ as equivalence classes of Cauchy sequences, not Dedekind’s cuts as outlined in this post.
I emphasise a natural way so that properties $(-1)\times (+\infty)=-\infty$ and $(-1)\times (-\infty)=+\infty$ follow as consequences.
 A: Sure - you can interpret $+\infty$ as the set $S_{+\infty}$ of all sequences of rationals whose limit is  $+\infty$ in the usual sense. That is, $$S_{+\infty}=\{(a_i)_{i\in\mathbb{N}}:a\in\mathbb{Q},  \forall x\exists m\forall n>m(a_n>x)\}.$$
Similarly, we can define $S_{-\infty}$ as the set of sequences of rationals whose limit is $-\infty$ in the usual sense.
Note that this exactly parallels the intuition behind the definition of reals as Cauchy sequences, where one informal slogan is "$a$ is the set of sequences of rationals converging to $a$."
It's then not hard to check that the usual definitions of arithmetic operations on equivalence classes of Cauchy sequences, when directly extended to $S_{+\infty}$ and $S_{-\infty}$, yield the desired results, while the intuitive undefinedness of $(+\infty)+(-\infty)$ is reflected in the fact that $S_{+\infty}+S_{-\infty}$ is not well-defined (different choices of representatives yield different results).

Let me say in detail why, for example, $S_{-\infty}+S_{+\infty}$ is (as hoped) undefined in this approach. Looking back at the standard reals, addition is defined as follows: for $A,B$ equivalence classes of Cauchy sequences we say $$A+B:=\{(a_i+b_i)_{i\in\mathbb{N}}: (a_i)_{i\in\mathbb{N}}\in A, (b_i)_{i\in\mathbb{N}}\in B\}.$$ We then prove that $A+B$ is in fact an equivalence class of Cauchy sequences; this isn't hard, but it is something we have to establish. 
We define $S_{-\infty}+S_{+\infty}$ in exactly the same way: as $$S_{-\infty}+S_{+\infty}:=\{(a_i+b_i)_{i\in\mathbb{N}}: (a_i)_{i\in\mathbb{N}}\in S_{-\infty}, (b_i)_{i\in\mathbb{N}}\in S_{+\infty}\}.$$ This is a perfectly meaningful object, but it's not an extended real number (in fact, it's just the set of all sequences of rationals - this is a good exercise).
So the usual definition of addition lifts without change to our slightly broader setting and behaves as expected.
A: Roughly speaking, the Cauchy definition identifies each real with the Cauchy sequences of rationals that converge to it, whereas the Dedekine definition identifies each real with the set of rational numbers less than it (or an ordered pair of that set, together with the set of greater rationals). Since you asked about the former, $\pm\infty$ is at first problematic because Cauchy sequences in $\Bbb Q$ don't $\to\infty$.
We'll have to consider something other than Cauchy sequences, but whatever sequences we do consider need to be split into equivalence classes by a suitable equivalence relation. The usual relation considers sequences equivalent if they differ by a null sequence. We can use this to partition sequences diverging to $\pm\infty$ if we really want. Do we use all of them, or just a subset that reminds us of Cauchy sequences? Well, since any sequence of rationals is Cauchy iff it has a real limit, we could have omitted "Cauchy" from the original definition anyway. So it seems natural to identify $\infty$ with the set of all sequences in $\Bbb Q$ that satisfy $\forall q\in\Bbb Q\exists N\in\Bbb N\forall n>N(a_n>q)$. We can handle $-\infty$ similarly, or just multiply the previous sequences by $-1$ elementwise.
