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!