Given any x ∈ℝ, show that there is a nonnegative integer k...

Given any $x \in \mathbb{R}$, show that there is a nonnegative integer $k$ and integers $a_{k}, a_{k-1}, \ldots, a_{1}, a_{0}, b_{1}, b_{2}, \ldots$ between 0 and 9 such that $$ x=\pm \lim _{n \rightarrow \infty}\left(a_{k} 10^{k}+a_{k-1} 10^{k-1}+\cdots+a_{0}+\frac{b_{1}}{10}+\frac{b_{2}}{10^{2}}+\cdots+\frac{b_{n}}{10^{n}}\right) . $$ (Hint: If $|x|<1$, set $k=0=a_{0}$ and apply Exercise 26 to $y:=|x|$, whereas if $|x| \geq 1$, apply Exercise 27 to $n:=[|x|]$ and Exercise 26 to $y:=|x|-n .)$ [Note: It is customary to call $a_{k}, a_{k-1}, \ldots, a_{0}, b_{1}, b_{2}, \ldots$ the digits of $x$ and write the above expression for $x$ as $x=\pm a_{k} a_{k-1} \ldots a_{0} \cdot b_{1} b_{2} \ldots$, and call it the decimal expansion of $x$.

Given any $x \in \mathbb{R}$, show that there is a nonnegative integer $k$ and integers $a_{k}, a_{k-1}, \ldots, a_{1}, a_{0}, b_{1}, b_{2}, \ldots$ between 0 and 9 such that
$$
x=\pm \lim _{n \rightarrow \infty}\left(a_{k} 10^{k}+a_{k-1} 10^{k-1}+\cdots+a_{0}+\frac{b_{1}}{10}+\frac{b_{2}}{10^{2}}+\cdots+\frac{b_{n}}{10^{n}}\right) .
$$
(Hint: If $|x|<1$, set $k=0=a_{0}$ and apply Exercise 26 to $y:=|x|$, whereas if $|x| \geq 1$, apply Exercise 27 to $n:=[|x|]$ and Exercise 26 to $y:=|x|-n .)$ [Note: It is customary to call $a_{k}, a_{k-1}, \ldots, a_{0}, b_{1}, b_{2}, \ldots$ the digits of $x$ and write the above expression for $x$ as $x=\pm a_{k} a_{k-1} \ldots a_{0} \cdot b_{1} b_{2} \ldots$, and call it the decimal expansion of $x$.

Key Concepts

Infinite Series and Convergence

The representation given in the statement is interpreted as the limit of a sequence of partial sums, forming an infinite series. Here, the idea of convergence is crucial because it ensures that as more digits are added to the expansion, the partial sum approaches the actual real number.

Decimal Expansion

A decimal expansion is a representation of a real number as a sum of digits multiplied by powers of 10. This shows how every number can be written in the base-10 numeral system, where each digit stands in a particular place (such as units, tens, hundreds, or fractional parts like tenths, hundredths, etc.) that determines its value contribution to the whole number.

Positional Number System

The positional number system is the method by which the value of a digit is determined by its place in the number. In the decimal system, each digit is multiplied by an appropriate power of 10 depending on its position, which is a fundamental concept for representing numbers in a compact and structured form.

Decomposition into Integer and Fractional Parts

Any real number can be decomposed into an integer part and a fractional part. This method allows one to separately handle the digits before and after the decimal point, ensuring that the entire real number is represented accurately by combining the expansion of its integer part (if any) with that of its fractional part.

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