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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.

Let x_n be a sequence of real numbers such that |x_n - x_{n+1}| ≤ 1/n for all n in N. Show that x_n is not a Cauchy sequence.

Explore the mathematical proof showing why a sequence with terms satisfying |x_n - x_{n+1}| ≤ 1/n is not a Cauchy sequence. This problem involves key concepts like the harmonic series and sequence convergence, suitable for undergraduate students.

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