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is the sequence a cauchy sequence if yes justify 

We are given a sequence \(\{p_n\}_{n=1}^{\infty}\) defined as follows:

- \(p_{2n+1} = 1\) for odd indices.
- \(p_{2n} = \frac{1}{n}\) for even indices.

The question asks whether this sequence is a Cauchy sequence. 

### Cauchy Sequence Definition:
A sequence \(\{p_n\}\) is called **Cauchy** if for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n > N\), the absolute difference \(|p_n - p_m|\) is less than \(\epsilon\). In other words, the terms of the sequence get arbitrarily close to each other as the sequence progresses.

### Analysis of the Sequence:

1. For **even indices** \(n=2k\), we have \(p_{2k} = \frac{1}{k}\), which converges to 0 as \(k \to \infty\).
2. For **odd indices** \(n = 2k+1\), we have \(p_{2k+1} = 1\), which is constant for all odd terms.

Thus, the sequence alternates between 1 (for odd indices) and values of \(\frac{1}{n}\) (for even indices). Since \(\frac{1}{n} \to 0\), the even-indexed terms approach 0, but the odd-indexed terms remain fixed at 1.

### Checking the Cauchy Condition:

Consider the difference between terms of the sequence:

- If we take an odd index \(p_{2k+1} = 1\) and an even index \(p_{2k} = \frac{1}{k}\), the difference between these two terms is:

\[
|p_{2k+1} - p_{2k}| = |1 - \frac{1}{k}|
\]

As \(k \to \infty\), this difference approaches 1, not 0.

Since the terms do not get arbitrarily close to each other (they maintain a difference of approximately 1 between odd and even terms), the sequence **does not satisfy** the condition for being Cauchy.

### Conclusion:
The sequence \(\{p_n\}\) is **not a Cauchy sequence** because the difference between the odd and even terms does not become arbitrarily small as \(n\) increases.

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Would you like further details or have any other questions?

Here are 5 related questions:
1. What is the formal definition of a convergent sequence?
2. How does a Cauchy sequence relate to convergence in a complete metric space?
3. Can a non-Cauchy sequence still be bounded?
4. What is an example of a sequence that is Cauchy but not convergent?
5. How does the behavior of even-indexed and odd-indexed terms affect the convergence of a sequence?

**Tip:** A sequence must get arbitrarily close to itself in order to be Cauchy. If the terms alternate or have large jumps, the sequence cannot be Cauchy.

Is the sequence {p_n} from the given definition Cauchy?

Explore the properties of a sequence {p_n} defined by alternating terms: odd terms set to 1 and even terms converging to 0. This detailed analysis checks whether the sequence meets the Cauchy condition, a critical concept in real analysis.

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