To solve these problems, let's go through each exercise one by one, showing that each sequence is a Cauchy sequence. A sequence $\{x_n\}$ is Cauchy if, for every $\epsilon > 0$, there exists an integer $N$ such that for all $m, n \geq N$, $|x_n - x_m| < \epsilon$.

1. Sequence $x_n = \frac{n}{n+1}$

To show that $\{x_n\}$ is Cauchy:

1. Calculate $|x_n - x_m|$: $x_n = \frac{n}{n+1} \quad \text{and} \quad x_m = \frac{m}{m+1}$ Then, $|x_n - x_m| = \left|\frac{n}{n+1} - \frac{m}{m+1}\right|$
2. Rewrite the expression with a common denominator and simplify.
3. Show that as $n, m \to \infty$, $|x_n - x_m|$ can be made arbitrarily small, proving that the sequence is Cauchy.

2. Sequence $x_n = 3 + \frac{1}{n}$

To show that $\{x_n\}$ is Cauchy:

1. Calculate $|x_n - x_m|$: $|x_n - x_m| = \left|(3 + \frac{1}{n}) - (3 + \frac{1}{m})\right| = \left|\frac{1}{n} - \frac{1}{m}\right|$
2. As $n, m \to \infty$, both $\frac{1}{n}$ and $\frac{1}{m}$ approach zero.
3. Show that $|x_n - x_m|$ can be made arbitrarily small for large $n, m$, which proves that the sequence is Cauchy.

3. Sequence $x_n = (-1)^n + \frac{1}{n}$

To determine if $\{x_n\}$ is Cauchy:

1. Calculate $|x_n - x_m|$: $|x_n - x_m| = \left|(-1)^n + \frac{1}{n} - \left((-1)^m + \frac{1}{m}\right)\right|$
2. This sequence does not converge, as it oscillates due to the $(-1)^n$ term.
3. Since the sequence does not converge, it is not Cauchy.

4. Sequence $x_n = \sqrt{n+1} - \sqrt{n}$

To show that $\{x_n\}$ is Cauchy:

1. Rewrite the expression: $x_n = \sqrt{n+1} - \sqrt{n} = \frac{1}{\sqrt{n+1} + \sqrt{n}}$
2. Calculate $|x_n - x_m|$ for large $n$ and $m$.
3. Show that as $n, m \to \infty$, the difference $|x_n - x_m|$ becomes arbitrarily small, proving that the sequence is Cauchy.

Would you like to see detailed steps for any specific part?

Further Related Questions:

1. What are the general properties of a Cauchy sequence?
2. How does the definition of Cauchy sequence relate to convergence?
3. What is the relationship between bounded sequences and Cauchy sequences?
4. Can an unbounded sequence be Cauchy? Explain with examples.
5. How does the presence of oscillatory terms affect the convergence of a sequence?

Tip: For a sequence to be Cauchy in $\mathbb{R}$, it must converge. Hence, checking if a sequence converges can simplify proving it's Cauchy.