Math Problem Statement
Solution
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 is decreasing and bounded below, then the sequence converges and:
"
Solution Overview:
To prove this, we can follow the steps below:
-
Monotonicity and Boundedness:
- The sequence is given as decreasing, meaning that for all .
- Additionally, it is bounded below, which means there exists a number such that for all .
-
Monotone Convergence Theorem (for real sequences):
- The monotone convergence theorem tells us that every monotone sequence that is bounded converges.
- Since is decreasing and bounded below, it must converge.
-
Infimum as the Limit:
- The limit of a decreasing sequence that is bounded below is the greatest lower bound (infimum) of the sequence. Thus:
Final Conclusion:
The sequence 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:
- What is the difference between monotonic and non-monotonic sequences?
- Can a non-bounded decreasing sequence converge?
- How does the Monotone Convergence Theorem apply to increasing sequences?
- What are other conditions that ensure the convergence of a sequence?
- 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.
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Math Problem Analysis
Mathematical Concepts
Monotonic Sequences
Convergence of Sequences
Infimum of Sets
Formulas
lim a_n = inf {a_n : n ∈ N}
Theorems
Monotone Convergence Theorem
Suitable Grade Level
Undergraduate - Introductory Analysis or Real Analysis
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