23 lines
878 B
TeX
23 lines
878 B
TeX
\section{Monotone Sequences}
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\begin{definition}
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Let $X=(x_n)$ be a sequence of real numbers. We say that $X$ is \textbf{increasing} if it satisfies the inequalities
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\[x_1 \leq x_2 \leq \dots \leq x_n \leq x_{n+1} \leq \dots\]
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We say that $X$ is \textbf{decreasing} if it satisfies the inequalities
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\[x_1 \geq x_2 \geq \dots \geq x_n \geq x_{n+1} \geq \dots\]
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We say that $X$ is \textbf{monotone} if it is either increasing or decreasing.
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\end{definition}
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\begin{theorem}[\textbf{Monotone Convergence Theorem}]
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A monotone sequence of real numbers is convergent if and only if it is bounded. Further:
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\begin{enumerate}
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\item If $X=(x_n)$ is a bounded increasing sequence, then
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\[\lim (x_n) = \sup \{x_n : n \in \N\}\]
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\item If $Y=(y_n)$ is a bounded decreasing sequence, then
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\[\lim (y_n) = \inf \{y_n : n \in \N\}\]
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\end{enumerate}
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\end{theorem}
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