In this answer to the question Is it true that $0.999999999\dots=1$?, Noah Snyder points out that

Symbols don't mean anything in particular until you've defined what you mean by them.

This question is motivated by the quote above. In real analysis, the symbol $0.99\cdots$ is defined as a real number, and $0.99\cdots=1$. One such definition can be seen in *Principles of Mathematical Analysis (3rd edition)* by Walter Rudin.^{1}

While $0.99\dots<1$ is a "false" intuition in the real number system, it is suggested by this Wikipedia article that the notion of "a number that falls short of $1$ by an infinitesimal amount" can be rigorously defined. See also this blog post by Terry Tao on *ultralimit analysis*.

Could anyone come up with an "introductory" reference (textbooks/papers) in nonstandard analysis **and** explain briefly how, in that reference, $0.99\cdots$ is defined in a way that is fundamentally different from the standard real analysis?

$\def\Rbf{\mathbf{R}}$

**Notes.**

In his Principles of Mathematical Analysis (3rd edition), Rudin introduced the set of

*real numbers*by the following theorem (page 8 of Chapter 1):There exists an ordered field $\mathbf{R}$ which has the least-upper-bound property. Moreover, $\mathbf{R}$ contains $\mathbf{Q}$ as a subfield.

In section 1.22, he what "decimal expansion" means for a given positive real number $x>0$ as follows.

Let $n_0$ be the largest integer such that $n_0\le x$. (The existence depends on the archimedean property of $\Rbf$.) Having chosen $n_0,n_1,\cdots,n_{k-1}$, let $n_k$ be the largest integer such that $$ n_0+\frac{n_1}{10}+\cdots+\frac{n_k}{10^k}\le x. $$ Let $E$ be the set of these number $$ n_0+\frac{n_1}{10}+\cdots+\frac{n_k}{10^k} \quad (k=0,1,2,\cdots).\tag{1} $$ Then $x=\sup E$. The decimal expansion of $x$ is $$ n_0\cdot n_1n_2n_3\cdots\tag{2} $$

**Conversely**, for any infinite decimal (2) the set $E$ of number (1) is bounded above, and (2) is the decimal expansion of $\sup E$.This gives one definition of $0.99\cdots$ by (1) and (2).