To prove that $\limsup(a_n + b_n) = a + b$ given that $(a_n)$ and $(b_n)$ are bounded sequences, $\lim a_n = a > 0$, and $\limsup b_n = b$, we'll proceed with the following steps:

Definitions and Assumptions

1. Bounded Sequences: Both $(a_n)$ and $(b_n)$ are bounded.
2. Convergence of $(a_n)$: $\lim a_n = a > 0$.
3. Limit Superior of $(b_n)$: $\limsup b_n = b$.

Proof

Step 1: Limiting Behavior of $(a_n)$

Since $\lim a_n = a > 0$, for any $\epsilon > 0$, there exists an $N_1$ such that for all $n > N_1$: $|a_n - a| < \epsilon.$ This implies: $a - \epsilon < a_n < a + \epsilon.$

Step 2: Upper Bound of $(b_n)$

By the definition of $\limsup b_n = b$, for any $\epsilon > 0$, there exists an $N_2$ such that for all $n > N_2$: $b_n < b + \epsilon.$ Also, there exist infinitely many indices $n$ such that $b_n > b - \epsilon$.

Step 3: Combining Limits for $(a_n + b_n)$

• Upper Bound: For $n > \max(N_1, N_2)$, $a_n + b_n < (a + \epsilon) + (b + \epsilon) = a + b + 2\epsilon.$ Therefore, $\limsup (a_n + b_n) \leq a + b + 2\epsilon.$ Since $\epsilon$ can be arbitrarily small, $\limsup (a_n + b_n) \leq a + b.$

• Lower Bound: Consider the infinitely many indices for which $b_n > b - \epsilon$ and combine them with the bounds for $a_n$: For these indices and $n > \max(N_1, N_2)$, $a_n + b_n > (a - \epsilon) + (b - \epsilon) = a + b - 2\epsilon.$ Therefore, $\limsup (a_n + b_n) \geq a + b - 2\epsilon.$

Since $\epsilon$ can be arbitrarily small, $\limsup (a_n + b_n) \geq a + b.$

Conclusion

From the inequalities, we have: $a + b \leq \limsup (a_n + b_n) \leq a + b.$

Thus, $\limsup (a_n + b_n) = a + b.$

This completes the proof.

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Would you like a more detailed breakdown of any part of the proof?

Here are some related questions for further exploration:

1. What is the difference between $\lim$ and $\limsup$?
2. How do you find the $\limsup$ of a sequence?
3. What is the significance of a sequence being bounded?
4. How does the value of $\lim a_n = a > 0$ affect the outcome of the problem?
5. What are the properties of bounded sequences?
6. Can the result still hold if $\lim a_n = 0$?
7. How can we generalize this proof to complex sequences?
8. What is the role of $\epsilon$ in the proof?

Tip: When working with limits and $\limsup$, always keep the definitions and properties in mind. This will help ensure accuracy and clarity in proofs.