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.

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.