The problem asks to show that the sequence $x_n$ is not a Cauchy sequence, given the condition:

$|x_n - x_{n+1}| \leq \frac{1}{n}, \quad \forall n \in \mathbb{N}.$

Solution Approach:

To determine whether $x_n$ is a Cauchy sequence, recall the definition of a Cauchy sequence: a sequence $\{x_n\}$ is Cauchy if for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $m, n \geq N$, we have:

$|x_n - x_m| < \epsilon.$

We need to show that the sequence $x_n$ does not satisfy this condition.

Step 1: Analyze the condition

We are given the condition:

$|x_n - x_{n+1}| \leq \frac{1}{n}.$

This means that the difference between consecutive terms of the sequence decreases as $n$ increases, but we still need to check if the cumulative difference between terms satisfies the Cauchy condition.

Step 2: Sum the differences between terms

Let’s estimate $|x_n - x_m|$ for $m > n$. Using the triangle inequality, we can write:

$|x_n - x_m| \leq |x_n - x_{n+1}| + |x_{n+1} - x_{n+2}| + \cdots + |x_{m-1} - x_m|.$

Using the given inequality, this becomes:

$|x_n - x_m| \leq \frac{1}{n} + \frac{1}{n+1} + \cdots + \frac{1}{m-1}.$

The right-hand side is a partial sum of the harmonic series, which is known to diverge as $m \to \infty$.

Step 3: Divergence of the Harmonic Series

The harmonic series $\sum \frac{1}{n}$ diverges, meaning that for large $n$ and $m$, the sum $\sum_{k=n}^{m-1} \frac{1}{k}$ can become arbitrarily large. In particular, it will not stay bounded for all $\epsilon > 0$, which is required for the sequence to be Cauchy.

Thus, for sufficiently large $n$ and $m$, we can make $|x_n - x_m|$ arbitrarily large, violating the condition for the sequence to be Cauchy.

Conclusion:

Since $|x_n - x_m|$ does not become arbitrarily small as $n \to \infty$, the sequence $\{x_n\}$ is not a Cauchy sequence.

Let me know if you'd like further details or clarifications.

Here are 5 questions you can consider based on this explanation:

1. What is the general definition of a Cauchy sequence?
2. Why does the harmonic series diverge, and how does that relate to this problem?
3. Could you provide an example of a sequence that is Cauchy?
4. What other conditions could we impose on a sequence to ensure that it is Cauchy?
5. How does the completeness of $\mathbb{R}$ relate to Cauchy sequences?

Tip: When dealing with sequences and limits, always analyze the sum of differences between terms to apply the Cauchy condition effectively.