https://file.aiquickdraw.com/mathbot/file/172736861965797ojdejj.jpg

Let's focus on solving **Question 6** and **Question 7** from the image.

### Question 6: Is the sequence \(\{h_n\}_{n=1}^{\infty}\) Cauchy?

Here, \( h_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \), which is known as the harmonic sequence. The harmonic sequence diverges to infinity as \(n \to \infty\), meaning it does not converge.

- **Cauchy Sequence Criterion**: A sequence \(\{h_n\}\) is Cauchy if for every \(\epsilon > 0\), there exists an integer \(N\) such that for all \(m, n \geq N\), \(|h_n - h_m| < \epsilon\).
  
However, since the harmonic sequence diverges, it cannot satisfy the Cauchy criterion. Thus, **the sequence \(\{h_n\}\) is not Cauchy**.

### Question 7: Is the sequence \(\{p_n\}_{n=1}^{\infty}\) Cauchy?

Here, we are given that:
- \( p_{2n+1} = 1 \)
- \( p_{2n} = \frac{1}{n} \)

This sequence alternates between \(1\) and \(\frac{1}{n}\) for even and odd indices. To check if this sequence is Cauchy, we need to analyze whether the difference between terms \(p_m\) and \(p_n\) can be made arbitrarily small as \(m, n \to \infty\).

1. For the odd terms, \( p_{2n+1} = 1 \), so they are constant.
2. For the even terms, \( p_{2n} = \frac{1}{n} \), which tends to \(0\) as \(n \to \infty\).

- The sequence alternates between a constant value of 1 and values that approach 0. Therefore, the difference between consecutive terms does not approach 0 as \(n \to \infty\).
  
Since the differences between terms do not tend to 0, **the sequence \(\{p_n\}\) is not Cauchy**.

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Would you like more details or clarification on any step?

Here are five related questions you might find useful:
1. What is the definition of a Cauchy sequence?
2. How does the divergence of the harmonic series affect its Cauchy property?
3. Can a non-convergent sequence ever be Cauchy?
4. What role does the behavior of alternating sequences play in determining if they are Cauchy?
5. Can an unbounded sequence be Cauchy?

**Tip**: A sequence that converges is always Cauchy, but not all Cauchy sequences converge unless they are in a complete space like the real numbers.

Explore whether the harmonic sequence h_n = 1 + 1/2 + 1/3 + ... + 1/n and the alternating sequence with terms p_{2n+1} = 1 and p_{2n} = 1/n are Cauchy sequences. This problem involves key mathematical concepts such as the harmonic series and Cauchy sequence criterion.

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