Show that if (an)n∈N,(bn)n∈N are equivalent sequences, then (an)n∈N is Cauchy if
and only if (bn)n∈N is Cauchy

To show that if \((a_n)_{n \in \mathbb{N}}\) and \((b_n)_{n \in \mathbb{N}}\) are equivalent sequences, then \((a_n)_{n \in \mathbb{N}}\) is Cauchy if and only if \((b_n)_{n \in \mathbb{N}}\) is Cauchy, we proceed as follows.

### Definitions:

1. **Equivalent Sequences**: Two sequences \((a_n)_{n \in \mathbb{N}}\) and \((b_n)_{n \in \mathbb{N}}\) are said to be equivalent if:
   \[
   \lim_{n \to \infty} |a_n - b_n| = 0
   \]
   This means that the difference between the terms of the sequences becomes arbitrarily small as \(n\) increases.

2. **Cauchy Sequence**: A sequence \((x_n)_{n \in \mathbb{N}}\) is called Cauchy if for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n \geq N\),
   \[
   |x_n - x_m| < \epsilon
   \]
   This means that the terms of the sequence get arbitrarily close to each other as \(n\) increases.

### Proof:

#### (⇒) If \((a_n)_{n \in \mathbb{N}}\) is Cauchy, then \((b_n)_{n \in \mathbb{N}}\) is Cauchy:

Assume \((a_n)_{n \in \mathbb{N}}\) is a Cauchy sequence. By the definition of a Cauchy sequence, for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for all \(m, n \geq N\),
\[
|a_n - a_m| < \frac{\epsilon}{3}
\]

Since \((a_n)_{n \in \mathbb{N}}\) and \((b_n)_{n \in \mathbb{N}}\) are equivalent, we know that:
\[
\lim_{n \to \infty} |a_n - b_n| = 0
\]
This means that there exists \(N_1 \in \mathbb{N}\) such that for all \(n \geq N_1\),
\[
|a_n - b_n| < \frac{\epsilon}{3}
\]
Similarly, for \(m \geq N_1\), we have
\[
|a_m - b_m| < \frac{\epsilon}{3}
\]

Now, for \(m, n \geq \max(N, N_1)\), we can use the triangle inequality to estimate \( |b_n - b_m| \):
\[
|b_n - b_m| \leq |b_n - a_n| + |a_n - a_m| + |a_m - b_m|
\]
Using the bounds derived above, we get:
\[
|b_n - b_m| \leq \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} = \epsilon
\]
Thus, \((b_n)_{n \in \mathbb{N}}\) is a Cauchy sequence.

#### (⇐) If \((b_n)_{n \in \mathbb{N}}\) is Cauchy, then \((a_n)_{n \in \mathbb{N}}\) is Cauchy:

Now assume that \((b_n)_{n \in \mathbb{N}}\) is a Cauchy sequence. By the definition of a Cauchy sequence, for every \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for all \(m, n \geq N\),
\[
|b_n - b_m| < \frac{\epsilon}{3}
\]

Since \((a_n)_{n \in \mathbb{N}}\) and \((b_n)_{n \in \mathbb{N}}\) are equivalent, there exists \(N_1 \in \mathbb{N}\) such that for all \(n \geq N_1\),
\[
|a_n - b_n| < \frac{\epsilon}{3}
\]
and for all \(m \geq N_1\),
\[
|a_m - b_m| < \frac{\epsilon}{3}
\]

Now, for \(m, n \geq \max(N, N_1)\), we can use the triangle inequality to estimate \( |a_n - a_m| \):
\[
|a_n - a_m| \leq |a_n - b_n| + |b_n - b_m| + |b_m - a_m|
\]
Using the bounds derived above, we get:
\[
|a_n - a_m| \leq \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} = \epsilon
\]
Thus, \((a_n)_{n \in \mathbb{N}}\) is a Cauchy sequence.

### Conclusion:

We have shown that if \((a_n)_{n \in \mathbb{N}}\) and \((b_n)_{n \in \mathbb{N}}\) are equivalent sequences, then \((a_n)_{n \in \mathbb{N}}\) is Cauchy if and only if \((b_n)_{n \in \mathbb{N}}\) is Cauchy.

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Would you like further clarification or more details?

Here are five related questions:
1. What is the significance of a Cauchy sequence in metric spaces?
2. How does the concept of equivalence of sequences relate to convergence?
3. What are some examples of equivalent sequences in the real number system?
4. Can a non-Cauchy sequence become Cauchy under some transformation?
5. What are some properties of sequences in non-Euclidean spaces?

**Tip**: In metric spaces, a sequence is Cauchy if and only if it converges when the space is complete.

Show that if (an)n∈N,(bn)n∈N are equivalent sequences, then (an)n∈N is Cauchy if and only if (bn)n∈N is Cauchy.

Explore the proof that if two sequences are equivalent, one is Cauchy if and only if the other is Cauchy. This detailed solution covers key concepts like Cauchy sequences, the definition of equivalent sequences, and uses the triangle inequality to establish the result. Suitable for undergraduate students studying real analysis or metric spaces.

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