Define $n(E):=\left\{\begin{matrix} \infty,& \text{if E is a not finite set} \\ |E|,& \text{if E is a finite set.} \end{matrix}\right.$
Let $\lbrace E_i\rbrace$ be an arbitrary sequence of disjoint sets of real numbers. I am wanting to show $n(\cup E_i)=\sum n(E_i)$ by using extended real numbers. [Note this problem is essentially the same as page 31, i.e. #4 (p. 42 of the pdf).]
I am going to as best I can fill in all the gaps for the lower limit sum relative to the index of the set.
Let $\lbrace E_i \rbrace_{i\in \mathbb{N}}$ be an arbitrary sequence of disjoint (pairwise) sets of real numbers.
I am again wanting to show $n(\bigcup\lbrace E_i \rbrace_{i \in \mathbb{N}})=\sum_{i \in \mathbb{N}}n(E_i)$.
However, this does not make sense to me as I could easily make up a function $E: \mathbb{N}\rightarrow \mathbb{R}$ where the correspondence is given by $E_j=\lbrace j \rbrace$, so then
\begin{align*}
\sum_{i \in \mathbb{N}}n(E_i)&=|E_1|+|E_2|+|E_3|+...\\
&=|\lbrace 1 \rbrace |+|\lbrace 2 \rbrace|+|\lbrace 3 \rbrace|+...\\
&=1+1+1+...\\
\end{align*}
which I thought would not be defined in this context. So what am I doing wrong here? Is the lower index of summation not supposed to be natural numbers $\mathbb{N}:=\lbrace 1, 2, 3, ... \rbrace$ or is the sequence part what I am messing up on? $\textbf{Question:}$ What am I doing wrong here?
You say you are confused as to why $$1+1+\dots=\infty.$$ An infinite sum is defined as the limit of the partial sums, and $$\lim_{n\to\infty}\sum_{i=1}^n1=\lim_{n\to\infty}n=\infty.$$
The sum is not well-defined if you only consider real numbers to be well defined. But this is the point of creating the extended real numbers in the first place. We want to be able to talk about $\infty$, so we formally add it to our number system.
What would be the point of having an $\infty$ symbol if you still had to say $1+1+\dots$ was undefined?
In the extended reals, we add the symbols $+\infty$ and $-\infty$, and we extend some of the structure of the reals (addition, multiplication, order, suprema, infinima and limits) to these symbols.
Specifically, just like a sequence of reals can have a real limit, a sequence of extended reals can have an extended real limit. The phrase "$\lim_{n\to\infty} a_n=L$", where $a_n$ and $L$ are extended reals, means the same thing when $L$ is a normal real number. When $L=+\infty$, we define that phrase to mean $$ \text{ for all } N>0 \text{ there exists } M \in \mathbb N:n\ge M\implies a_n \ge N. $$ Using this definition, it is easy to prove $1+1+\dots=\infty$ by proving the limit of the partial sums is infinity.
If you cannot accept this explanation, you need to find an authority who can define what limits mean for the extended real numbers.