Proof
(related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
 Assume, the real sequence
$\((a_n)_{n\in\mathbb N}\)
is monotonic and bounded.
 According to the theorem of BolzanoWeierstrass, it has a convergent subsequence \((a_{n_k})_{k\in\mathbb N}\) with\[\lim_{n\rightarrow\infty} a_{n_k}=a.\]
 According to the definition of convergence, this means that for every \(\epsilon > 0\) there is an index \(K(\epsilon)\in\mathbb N\) with \[a_{n_k}  a < \epsilon\quad\quad\forall k \ge K(\epsilon).\]
 Set \(N(\epsilon):=n_{K(\epsilon)}\). For every \(n > N(\epsilon)\) there is a \(k\ge K(\epsilon)\) with
\[ n_k\le n < n_{k+1}\quad\quad( * ).\]
 Since \((a_n)_{n\in\mathbb N}\) is monotonically increasing (or monotonically decreasing), it follows from \( ( * ) \) that \[ a_{n_k}\le a_n \le a_{n_{k+1}}\quad\quad\text{or}\quad\quad a_{n_k}\ge a_n \ge a_{n_{k+1}}.\]
 In either case we have for all \(n > N(\epsilon)\):
\[a_n  a\le\max(a_{n_k}  a,a_{n_{k+1}}  a) < \epsilon.\]
 This shows that \((a_n)_{n\in\mathbb N}\) is convergent.
∎
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983