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

To prove that \(\limsup (x_n + y_n) \leq \limsup (x_n) + \limsup (y_n)\), let's proceed step by step.

### Step 1: Definition of \(\limsup\)

For a bounded sequence \((x_n)\), the \(\limsup(x_n)\) is defined as:
\[
\limsup(x_n) = \lim_{n \to \infty} \sup_{k \geq n} x_k
\]
Similarly, for the sequence \((y_n)\),
\[
\limsup(y_n) = \lim_{n \to \infty} \sup_{k \geq n} y_k
\]
And for the sequence \((x_n + y_n)\),
\[
\limsup(x_n + y_n) = \lim_{n \to \infty} \sup_{k \geq n} (x_k + y_k)
\]

### Step 2: Supremum of the Sum

The supremum of the sum of two sequences satisfies the following inequality:
\[
\sup_{k \geq n} (x_k + y_k) \leq \sup_{k \geq n} x_k + \sup_{k \geq n} y_k
\]

This inequality holds because, for each \(k\),
\[
x_k + y_k \leq \sup_{k \geq n} x_k + \sup_{k \geq n} y_k
\]
and thus,
\[
\sup_{k \geq n} (x_k + y_k) \leq \sup_{k \geq n} x_k + \sup_{k \geq n} y_k
\]

### Step 3: Taking the Limit Superior

Now, taking the limit superior as \(n \to \infty\) on both sides of the inequality above, 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 the limit superior of the sum is less than or equal to the sum of the limit superiors:
\[
\limsup (x_n + y_n) \leq \limsup(x_n) + \limsup(y_n)
\]

### Conclusion

Thus, we've shown that:
\[
\limsup(x_n + y_n) \leq \limsup(x_n) + \limsup(y_n)
\]

This concludes the proof.

Do you want more details or have any questions?

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Here are 8 related questions to further your understanding:

1. What is the difference between \(\limsup\) and \(\liminf\)?
2. Can \(\limsup(x_n + y_n)\) be strictly less than \(\limsup(x_n) + \limsup(y_n)\)?
3. How would you prove a similar inequality for \(\liminf(x_n + y_n)\)?
4. What happens if one of the sequences \((x_n)\) or \((y_n)\) is unbounded?
5. How does this inequality relate to the sum of two convergent sequences?
6. Can you give an example of sequences where the inequality is strict?
7. How do \(\limsup\) and \(\liminf\) behave under multiplication of two sequences?
8. Can you extend this proof to more than two sequences?

**Tip:** Understanding the behavior of \(\limsup\) and \(\liminf\) is crucial in real analysis, especially when dealing with sequences that do not converge.

Learn how to prove the inequality \(\limsup (x_n + y_n) \leq \limsup (x_n) + \limsup (y_n)\) for bounded sequences \((x_n)\) and \((y_n)\). This proof involves concepts from real analysis, including the limit superior of sequences and the properties of bounded sequences.

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