= Bounded monotone sequence theorem
= Monotone convergence theorem for sequences
{synonym}
\b[A <monotone sequence> of <real numbers> has a finite <limit of a sequence> exactly when it is a <bounded sequence>.] In the increasing case, only an <upper bound> needs to be supplied, because the first term already gives a <lower bound>; in the decreasing case, only a <lower bound> needs to be supplied. The <limit of a sequence> is the exact edge of the <set> of terms:
$$
a_n\leq a_{n+1}\leq M\quad\Longrightarrow\quad a_n\longrightarrow L=\sup\{a_n:n\geq1\},
$$
$$
m\leq b_{n+1}\leq b_n\quad\Longrightarrow\quad b_n\longrightarrow L=\inf\{b_n:n\geq1\}.
$$
The <upper bound> $M$ or <lower bound> $m$ need not itself be the <limit of a sequence>. A <supremum> or <infimum> identifies the sharp bound. <Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl>[§2.1.1, Theorem 2.1.10 and the one-sided-boundedness observation following its proof]{ref}
For example, $a_n=1-1/n$ is increasing and bounded above by one, and $b_n=1+1/n$ is decreasing and bounded below by one. Both have <limit of a sequence> one, although neither ever equals one. Their <absolute errors> are exactly $1/n$. They also bracket any <sequence> $x_n$ satisfying $a_n\leq x_n\leq b_n$: the <squeeze theorem> gives $x_n\to1$, even if $(x_n)$ itself is not a <monotone sequence>.
\Image[/bounded-monotone-sequence-limit-brackets.png]
{title=Increasing and decreasing <sequences> bracket a common <limit of a sequence>, with explicitly shrinking <absolute errors>}
{description=Left: $1-1/n$ and $1+1/n$ enclose one; for every $n\geq6$ both lie strictly inside the tolerance band $1\pm0.2$. Right: the <absolute errors> $1/n$ and $2^{-n}$ shrink at different rates. The markers are individual terms, and connecting segments only guide the eye.}
The theorem is useful when a <recurrence relation> gives successive improvements and a bound, before the answer has been identified. It proves existence of a <limit of a sequence>; further algebra, often using <continuity>, identifies the answer. A finite plot illustrates the formulas but does not establish the infinite claim. <Error bounds for bounded monotone sequences> require additional quantitative information.
A real increasing <sequence> bounded above converges to its <supremum>; a decreasing <sequence> bounded below converges to its <infimum>. For the increasing case, the definition of <supremum> provides a term above $L-\varepsilon$, and every later term remains between that term and $L$. Negation reduces the decreasing case to the increasing case. This elementary result is distinct from the measure-theoretic <monotone convergence theorem> for integrals.
The existence of the real <supremum> in this short proof uses the <least-upper-bound property>, equivalently <completeness of the real numbers>. The <approximation property of a supremum> supplies $a_N>L-\varepsilon$, and monotonicity keeps every later term within the same tolerance:
$$
n\geq N\quad\Longrightarrow\quad 0\leq L-a_n\leq L-a_N<\varepsilon.
$$
The corresponding decreasing statement follows from the <approximation property of an infimum>. The converse uses the fact that every <convergent sequence> is a <bounded sequence>. <Basic Analysis: Introduction to Real Analysis, Volume I by Jiří Lebl>[§2.1, Proposition 2.1.7 and §2.1.1, proof of Theorem 2.1.10]{ref}
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