42 lines
1.2 KiB
TeX
42 lines
1.2 KiB
TeX
\section{Absolute Value and the Real Line}
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\begin{definition}
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The \textbf{absolute value} of a real number $a$, denoted by $|a|$, is defined by
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\[|a|:=\begin{cases}
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a & \text{if } a>0, \\
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0 & \text{if } a=0, \\
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-a & \text{if } a<0.
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\end{cases}\]
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\end{definition}
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\begin{theorem}
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\begin{enumerate}
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\item[]
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\item $|ab|=|a||b|$ for all $a,b \in \R$.
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\item $|a|^2 = a^2$ for all $a \in \R$.
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\item If $c \geq 0$, then $|a| \leq c$ if and only if $-c \leq a \leq c$.
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\item $-|a|\leq a \leq |a|$ for all $a \in \R$.
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\end{enumerate}
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\end{theorem}
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\begin{theorem}[\textbf{Triangle Inequality}]
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If $a,b \in \R$, then $|a+b| \leq |a| + |b|$.
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\end{theorem}
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\begin{corollary}
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If $a,b \in \R$, then
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\begin{enumerate}
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\item $\left| |a|-|b| \right| \leq |a-b|$,
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\item $|a-b| \leq |a| + |b|$.
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\end{enumerate}
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\end{corollary}
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\begin{corollary}
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If $a_1, a_2, \dots, a_n$ are any real numbers, then
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\[|a_1 + a_2 + \dots + a_n| \leq |a_1| + |a_2| + \dots + |a_n|\]
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\end{corollary}
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\begin{definition}
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Let $a \in \R$ and $\varepsilon > 0$. Then the $\varepsilon$-\textbf{neighborhood} of $a$ is the set $V_\varepsilon(a):=\{x \in \R : |x-a| < \varepsilon\}$.
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\end{definition}
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