https://file.aiquickdraw.com/mathbot/file/1727668203393qjzv2nf5.jpg

The problem presented in the image is written in Indonesian and appears to be about proving a property of a sequence of real numbers. Here's a translation of the key parts of the problem:

---
**Problem:**

"Prove that if the sequence \((a_n)\) is decreasing and bounded below, then the sequence \((a_n)\) converges and:

\[
\lim_{n \to \infty} a_n = \inf \{a_n : n \in \mathbb{N}\}
\]
"

---

### Solution Overview:

To prove this, we can follow the steps below:

1. **Monotonicity and Boundedness:**
   - The sequence \((a_n)\) is given as decreasing, meaning that \(a_{n+1} \leq a_n\) for all \(n\).
   - Additionally, it is bounded below, which means there exists a number \(m\) such that \(a_n \geq m\) for all \(n\).

2. **Monotone Convergence Theorem (for real sequences):**
   - The monotone convergence theorem tells us that every monotone sequence that is bounded converges.
   - Since \((a_n)\) is decreasing and bounded below, it must converge.

3. **Infimum as the Limit:**
   - The limit of a decreasing sequence that is bounded below is the greatest lower bound (infimum) of the sequence. Thus:
   \[
   \lim_{n \to \infty} a_n = \inf \{a_n : n \in \mathbb{N}\}.
   \]

### Final Conclusion:
The sequence \((a_n)\) converges, and its limit is the infimum of the terms of the sequence.

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Let me know if you'd like further clarifications or additional details.

Here are 5 related questions:
1. What is the difference between monotonic and non-monotonic sequences?
2. Can a non-bounded decreasing sequence converge?
3. How does the Monotone Convergence Theorem apply to increasing sequences?
4. What are other conditions that ensure the convergence of a sequence?
5. How can we determine if a sequence is bounded?

**Tip:** In general, when working with sequences, always check if they are monotonic (increasing or decreasing) and whether they are bounded. These properties often help in determining convergence.

Buktikan jika barisan (a_n) turun monoton dan terbatas ke bawah, maka barisan (a_n) konvergen dan lim a_n = inf {a_n : n ∈ N}

Learn how to prove that a decreasing and bounded sequence of real numbers converges. This detailed solution involves the Monotone Convergence Theorem and the concept of the infimum of a set, suitable for undergraduate students studying real analysis.

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