A sequence $(s_n) $ of real numbers is called an increasing sequence if $s_n \leq s_{n+1} $ for all $n $, and $(s_n) $ is called a decreasing sequence if $s_n \geq s_{n+1} $ for all $n $. Note that if $(s_n) $ is increasing, then $s_n \leq s_m$ whenever $n < m$. A sequence that is increasing or decreasing will be called a monotone sequence or a monotonic sequence.
All bounded monotone sequences converge.
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If $(s_n)$ is an unbounded increasing sequence, then $\lim s_n = +\infty $. If $(s_n)$ is an unbounded decreasing sequence, then $\lim s_n = -\infty $.
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If $s_n $ is a monotone sequence, then the sequence either converges, diverges to $+\infty$ or diverges to $-\infty $. Thus $\lim s_n $ always exists for monotone sequences.
Let $(s_n) $ be a sequence in $\R $. We define
$\lim \supr{ s_n } = \limu{ N }{ \infty } \supr{ \set{ s_n : n > N }} $
$\lim \infi{ s_n } = \limu{ N }{ \infty } \infi{ \set{ s_n : n > N }} $
In this definition we do not restrict $(s_n) $ to be bounded. However, we adopt the following conventions. If $(s_n) $ is not bounded above, $\supr{ s_n : n > N } = +\infty$ for all $N $ and we decree $\lim \supr{ s_n } = +\infty$. Likewise, if $(s_n) $ is not bounded below, $\infi{ \set{ s_n : n > N }} = -\infty $ for all $N $ and we decree $\lim \infi{ s_n } = -\infty $.
$\lim \supr{ s_n }$ may not equal $\supr{ s_n : n \in \N } $, but $\lim \supr{ s_n } \leq \supr{ \set{ s_n : n \in \N }}$.
Let $(s_n) $ be a sequence in $\R $.
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If $\lim s_n $ exists, then $\lim \infi{ s_n } = \lim s_n = \lim \supr{ s_n } $.
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If $\lim s_n = \lim \supr{ s_n }$, then $\lim s_n$ exists and $\lim s_n = \lim \infi{ s_n } = \lim \supr{ s_n } $.
A sequence $(s_n) $ of real numbers is called a Cauchy sequence if
for each $\epsilon > 0 $ there exists a number $N $ such that
$$m,n > N \implies \abs{ s_n - s_m } < \epsilon $$
Convergent sequences are Cauchy sequences.
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Cauchy sequences are bounded.
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Let $(s_n) $ be a sequence in $\R $.
$(s_n)$ is a convergent sequence if and only if it is a Cauchy sequence.
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