To show that a convergent sequence $(a_n)$ in $\mathbb{R}$ is also a Cauchy sequence, we begin by recalling the definitions of both concepts:

Convergent Sequence:

A sequence $(a_n)$ in $\mathbb{R}$ is said to converge to a limit $L \in \mathbb{R}$ if for every $\epsilon > 0$, there exists an index $N \in \mathbb{N}$ such that for all $n \geq N$, $|a_n - L| < \epsilon.$

Cauchy Sequence:

A sequence $(a_n)$ in $\mathbb{R}$ is called a Cauchy sequence if for every $\epsilon > 0$, there exists an index $N \in \mathbb{N}$ such that for all $m, n \geq N$, $|a_n - a_m| < \epsilon.$

Proof that a Convergent Sequence is Cauchy:

Assume that $(a_n)$ converges to some limit $L$, i.e., for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$, $|a_n - L| < \frac{\epsilon}{2}.$

Now, for any $m, n \geq N$, we can apply the triangle inequality: $|a_n - a_m| = |a_n - L + L - a_m| \leq |a_n - L| + |a_m - L|.$ By the assumption of convergence, both $|a_n - L|$ and $|a_m - L|$ are each less than $\frac{\epsilon}{2}$ for $n, m \geq N$. Therefore, $|a_n - a_m| \leq |a_n - L| + |a_m - L| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon.$

Thus, for all $m, n \geq N$, we have $|a_n - a_m| < \epsilon$, which shows that $(a_n)$ is a Cauchy sequence.

Conclusion:

Every convergent sequence in $\mathbb{R}$ is a Cauchy sequence.

---

Do you want further details on any step of this proof or have any questions? Here are five related questions to deepen your understanding:

1. What is the difference between a Cauchy sequence and a convergent sequence?
2. Can a sequence be Cauchy but not convergent in spaces other than $\mathbb{R}$?
3. How does completeness of $\mathbb{R}$ relate to this proof?
4. Can you construct an example of a convergent sequence that is also Cauchy?
5. How would the proof differ if we were dealing with complex sequences?

Tip: In a complete metric space like $\mathbb{R}$, every Cauchy sequence converges. This property is key in many advanced mathematical concepts.