While studying measure theory, one encounters inevitably the extended real numbers $\overline{\mathbb{R}} := \mathbb{R} \cup \{-\infty, + \infty\}$.
I would like a book/reference that properly defines doing analysis with this stuff.
(1) For example, let $(a_n)_{n=1}^\infty$ be a sequence in $\mathbb{R}^{+}\cup\{+\infty\}$.
Then, it is intuitively clear that $\sum_{n=1}^\infty a_n = \infty$ when there is $m \geq 1 $ s.t. $a_m = +\infty$.
I want a book that defines what $\sum_{n=1}^\infty a_n $ means where one of the terms can be infinity. Is this done by putting a metric on the extended real numbers? Is this done by declaring something like $a + \infty = \infty$ for all $a \in \mathbb{R}$? The book should definitely answer these questions.
(2) More general, the book should define what $\lim_{n \to \infty} a_n$ means where $(a_n)$ is a sequence of extended real numbers (i.e. one of the terms can be infinity).
Best Answer
Berberian's book Fundamentals of real analysis defines
$a+\infty$ in page 74.
$\displaystyle\sum_{n=1}^\infty a_n$ where one of the terms can be infinity in page 76.
$\displaystyle\lim_{n\to\infty} a_n$ where $(a_n)$ is a sequence of extended real numbers in page 82.