if (an) and (bn) are bounded sequences and liman=a>0 and limsupbn=b then show that limsup(an+bn)=a+b

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.

---

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.

Explore the proof that limsup(an + bn) = a + b for bounded sequences where lim an = a > 0 and limsup bn = b. Understand the concepts of sequences, limits, and lim sup in mathematics. Suitable for college-level mathematics students.

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