Neighborhoods in 𝐑

The Real Number Line

One way to represent the real numbers $\mathbb {R}$ is on the real number line as depicted below.

Real number line

We will now state the important geometric representation of the absolute value with respect to the real number line.

Definition: If $a$ and $b$ are real numbers, then we say that the distance from $a$ to the origin $0$ is the absolute value of $a$, $\mid a\mid$. We say that the distance between $a$ and $b$ is the absolute value of their difference, namely $\mid a-b\mid$.

For example consider the numbers $-2$ and $2$. There is a distance of $4$ in between these numbers because $\mid -2-2\mid =\mid -4\mid =4$.

Epsilon Neighbourhood of a Real Number

Definition: Let $a$ be a real number and let $\varepsilon >0$. The $\varepsilon$-neighbourhood of the number $a$ is the set denoted $V_{\varepsilon }(a):=\{x\in \mathbb {R} :\mid x-a\mid <\varepsilon \}$. Alternatively we can define $V_{\varepsilon }(a):=\{x\in \mathbb {R} :a-\varepsilon <x<a+\varepsilon \}$. $\varepsilon$-neighbourhood around $a$ ($V_{\varepsilon }(a)$) expressed on the real number line

For example, consider the point $1$, and let $\varepsilon _{0}=2$. Then $V_{\varepsilon _{0}}(1)=\{x\in \mathbb {R} :\mid x-1\mid <2\}=(-1,3)$.

We will now look at a simple theorem regarding the epsilon-neighbourhood of a real number.

Theorem 1: Let $a$ be a real number. If $\forall \varepsilon >0$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in V_{\varepsilon} (a)} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a} .

• Proof of Theorem 1: Suppose that for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall \varepsilon > 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid x - a \mid < \varepsilon} . We know that then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid x - a \mid = 0} if and only if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x - a = 0} and therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = a} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}

Resources

• The Real Line and The Epsilon Neighbourhood of a Real Number, mathonline.wikidot.com