Suppose the sequence (an)n∈N over R is convergent. Show that it is also Cauchy.

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.

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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.

Suppose the sequence (a_n)_n∈N over R is convergent. Show that it is also Cauchy.

Explore the proof that every convergent sequence in the real numbers is also a Cauchy sequence. This problem involves fundamental concepts from real analysis, such as sequence convergence, the definition of Cauchy sequences, and the triangle inequality. Suitable for university-level mathematics students.

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