let (xn) and (yn) be bounded sequences then prove that limsup(xn + yn) <= lim (xn) + lim (yn)

Let's prove the inequality \(\limsup(x_n + y_n) \leq \limsup(x_n) + \limsup(y_n)\).

### Definitions and Key Points:
1. **Bounded Sequences:** A sequence \((x_n)\) is bounded if there exists a real number \(M > 0\) such that \(|x_n| \leq M\) for all \(n\).
2. **\(\limsup\):** The limit superior (or \(\limsup\)) of a sequence \((x_n)\) is defined as:
   \[
   \limsup x_n = \lim_{n \to \infty} \sup_{k \geq n} x_k
   \]
   Essentially, it represents the largest value that the sequence approaches infinitely often.

### Proof:
Let's denote \(\alpha = \limsup x_n\) and \(\beta = \limsup y_n\). By definition, we have:
\[
\alpha = \lim_{n \to \infty} \sup_{k \geq n} x_k \quad \text{and} \quad \beta = \lim_{n \to \infty} \sup_{k \geq n} y_k
\]

Now, consider the sequence \(z_n = x_n + y_n\). The \(\limsup\) of \(z_n\) is:
\[
\limsup z_n = \limsup (x_n + y_n)
\]

According to the properties of \(\sup\) and \(\limsup\):
\[
\sup_{k \geq n} (x_k + y_k) \leq \sup_{k \geq n} x_k + \sup_{k \geq n} y_k
\]
Taking the \(\limsup\) on both sides, we get:
\[
\limsup_{n \to \infty} \sup_{k \geq n} (x_k + y_k) \leq \limsup_{n \to \infty} \left(\sup_{k \geq n} x_k + \sup_{k \geq n} y_k\right)
\]

Since \(\sup_{k \geq n} x_k \to \alpha\) and \(\sup_{k \geq n} y_k \to \beta\), we conclude:
\[
\limsup (x_n + y_n) \leq \alpha + \beta
\]

Therefore:
\[
\limsup (x_n + y_n) \leq \limsup x_n + \limsup y_n
\]
which is the desired inequality.

### Conclusion:
We have shown that the limit superior of the sum of two bounded sequences is less than or equal to the sum of their limit superiors.

---

Would you like further details on any part of this proof or have any questions?

Here are some related questions to consider:

1. What is the difference between \(\limsup\) and \(\liminf\)?
2. How does this proof change if one of the sequences is unbounded?
3. What are the implications of this result in terms of convergence?
4. Can we say anything about \(\liminf(x_n + y_n)\) in relation to \(\liminf x_n\) and \(\liminf y_n\)?
5. How does the result change if \(x_n\) and \(y_n\) are both convergent sequences?
6. What are some real-world applications of limit superior and limit inferior?
7. How can we use this inequality to bound the behavior of more complex sequences?
8. Can the equality \(\limsup(x_n + y_n) = \limsup(x_n) + \limsup(y_n)\) ever hold, and under what conditions?

**Tip:** Understanding the concepts of \(\limsup\) and \(\liminf\) can provide deep insights into the long-term behavior of sequences, especially in cases where sequences do not converge.

Explore the proof that \(\limsup(x_n + y_n) \leq \limsup(x_n) + \limsup(y_n)\). Understand the concepts of \(\limsup\), bounded sequences, and inequalities in mathematical sequences. Suitable for advanced undergraduate students and researchers.

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