2.2 Absolute Value and the Real Line

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2.2.1 Definition: Absolute value

The absolute value $a\in \mathbb{R}$, denoted by $|a|$, is defined by $|a|:=\{\begin{array}{ll}-a,& a<0\\ 0,& a=0\\ a,& a>0\end{array}$

We see from the definition of absolute value that, $\forall a\in \mathbb{R}$,

• $|a|\ge 0$
• $|-a|=|a|$
• $|a|=0⟺a=0$

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2.2.2 Theorem: Additional property of absolute values

$\forall a,b,c\in \mathbb{R},c\ge 0:$

1. $|ab|=|a||b|$
2. $|a{|}^2=a^2$
3. $|a|\le c⟺-c\le a\le c$
4. $-|a|\le a\le |a|$

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2.2.3 Theorem: Triangle inequality

$|a+b|\le |a|+|b|,\forall a,b\in \mathbb{R}$

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2.2.4 Corollary: From 2.2.3

There are multiple variations of Triangle Inequality, which is:

$\forall a,b\in \mathbb{R}:$

1. $||a|-|b||\le |a-b|$
2. $|a-b|\le |a|+|b|$

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2.2.5 Corollary: From 2.2.3

$|a_1+\cdots +a_n|\le |a_1|+\cdots +|a_n|,\forall a_1,\dots ,a_n\in \mathbb{R}$

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2.2.7 Definition: Neighborhood

Definition

Let $a\in \mathbb{R}$ and $\epsilon >0$. Then the $\epsilon$-neighborhood of $a$ is the set $V_{\epsilon }(a):=\{x\in \mathbb{R}:\mid x-a|<\epsilon \}$

Remark

This essentially means that $x$ is “close” to $a$. $V_{\epsilon }(a)$ is the set of all real number within $\epsilon$ unit “distance” of $a$.

For example 2-neighborhood of 3 means numbers within 2 unit distance of 3, which ranges from 1 to 5.

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2.2.8 Theorem: Uniqueness of a point in neighborhoods

Theorem

Let $a\in \mathbb{R}$.

Then $x\in V_{\epsilon }(a)⟹x=a,\forall \epsilon >0$

Remark

We can make sense of this theorem by imagining even if $\epsilon$ (or the distance between $x$ and $a$) is infinitesimally small, but $x$ is still in $V_{\epsilon }(a)$.

This can only happen if the distance between $x$ and $a$ is $0$, which means $x=a$.

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