Proof

Assume, the real sequence $$(a_n)_{n\in\mathbb N}$ is monotonic and bounded.
According to the theorem of Bolzano-Weierstrass, 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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Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983