Proof
(related to Proposition: Convergent Real Sequences Are Cauchy Sequences)
- By hypothesis, a given real sequence $(a_n)_{n\in\mathbb N}$ is convergent to the limit $a\in\mathbb R.$
- This means that for a given $\epsilon > 0$ there is an index $N\in\mathbb N$ such that $$|a_n-a| < \frac{\epsilon}2$$ for all $n\ge N.$
- Thus, by the triangle inequality $$\begin{align}|a_n- a_m|&=|a_n-a+a-a_m|\nonumber\\
&\le |a_n-a|+|a-a_m|\nonumber\\
&\le \frac{\epsilon}{2}+\frac{\epsilon}{2}\nonumber\\
&=\epsilon\nonumber\end{align}$$ for all $n,m\ge N.$
- It follows that $(a_n)_{n\in\mathbb N}$ is a real Cauchy sequence.
∎
Thank you to the contributors under CC BY-SA 4.0! 
- Github:
-
References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
