If a 
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  converges, then {a 
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The problem states that \( \{ a_n \} \) is a sequence such that \( a_n \geq 1 \) and the sequence \( a_n + a_{n+1} \) converges. We are asked to show that \( \{ a_n \} \) is also convergent.

### Approach:
We know that the sequence \( a_n + a_{n+1} \) converges. Let the limit of this sequence be \( L \), i.e.,

\[
\lim_{n \to \infty} (a_n + a_{n+1}) = L
\]

This implies that, for large \( n \), the difference between \( a_n \) and \( a_{n+1} \) must get smaller as \( n \to \infty \). Specifically:

\[
\lim_{n \to \infty} (a_{n+1} - a_n) = 0
\]

This suggests that the sequence \( \{ a_n \} \) is becoming closer to a constant value as \( n \) increases. Since the difference between consecutive terms of the sequence approaches zero, the sequence \( \{ a_n \} \) is either bounded or converges to some limit.

### Proof Outline:

1. **Boundedness**: The sequence \( \{ a_n \geq 1 \} \) is bounded below by 1. If \( a_n + a_{n+1} \to L \), then neither \( a_n \) nor \( a_{n+1} \) can grow indefinitely, so \( \{ a_n \} \) must be bounded above as well. Thus, \( \{ a_n \} \) is bounded.

2. **Cauchy sequence**: Since \( \lim_{n \to \infty} (a_{n+1} - a_n) = 0 \), the sequence behaves like a Cauchy sequence. In a real-numbered sequence, a Cauchy sequence converges if it is bounded.

Thus, by the properties of real numbers, \( \{ a_n \} \) converges.

### Conclusion:
Since \( a_n + a_{n+1} \) converges, the sequence \( \{ a_n \} \) is also convergent.

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Would you like more details on any part of the explanation?

Here are 5 related questions you could consider:

1. What does it mean for a sequence to be bounded and why is that important for convergence?
2. Can you show that the condition \( \lim_{n \to \infty} (a_{n+1} - a_n) = 0 \) implies that the sequence is Cauchy?
3. How do Cauchy sequences guarantee convergence in the real numbers?
4. Can this method be applied to sequences where \( a_n \) is not bounded below by 1?
5. What happens if the sequence \( a_n + a_{n+1} \) diverges?

**Tip**: A sequence is convergent if and only if it is both bounded and satisfies the Cauchy criterion!

If a_n ≥1 and a_n + a_(n+1) converges, then {a_n} is convergent.

Explore the proof that if a sequence {a_n} satisfies a_n ≥ 1 and a_n + a_(n+1) converges, then {a_n} is also convergent. This problem highlights key concepts in real analysis, including boundedness, Cauchy sequences, and the Cauchy Convergence Theorem.

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