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.

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.