# 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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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983