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.