In daily life, it is common practice to use a sequence of number elements as an integer, e.g., 999 is a decimal number. As I am reading rigorous construction of numbers from Rudin's mathematical analysis, I would like to find a rigorous explanation of such uses. To simplify the problem, I will focus on the case of binary numbers, as $0$ and $1$ are already defined together with $+$ and $\cdot$ in the construction of integers. Introducing $2$, $3$, etc. is unnecessary.
My definition goes as follows. First of all, any daily use of number representation is a sequence. For example, 8-bit binary $01110011$ is a sequence $a$ such that $a_{0} = 1$, $a_{1}= 1$, $a_{2} = 0$, $a_{3} = 0$, $a_{4} = 1$, $a_{5} = 1$, $a_{6} = 1$ and $a_{7} = 0$ and for any $n > 7$, $a_{n} = 0$. Then for such sequence $a$, we define a sequence $c$ for weights: \begin{equation} c_{0} = 1 \end{equation} and for any $n \in \mathbb{N}$, \begin{equation} c_{n + 1} = (1 + 1) \cdot c_{n}. \end{equation} Notice that I avoided using $2$ on purpose for rigor. Further, we define a sequence $b$: \begin{equation} b_{0} = c_{0} \cdot a_{0} = a_{0} \end{equation} and for any $n \in \mathbb{N}$ \begin{equation} b_{n + 1} = b_{n} + c_{n + 1} \cdot a_{n + 1}. \end{equation} The interpretation for $a$ as an integer is then $\lim_{n \to \infty}b_{n}$. Note that essentially, I constructed a function $f$ that maps each representation $a: \mathbb{N} \to \left\{0,1\right\}$ to some $s \in \mathbb{Z}$. Is this a rigorous interpretation?
Another question, that may not be relevant to math directly: why is compilation of latex/mathjax on this website so fast? Nearly real time. Much better than overleaf. Can someone explain :)