Math Problem Statement

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}

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 (an)(a_n) is decreasing and bounded below, then the sequence (an)(a_n) converges and:

limnan=inf{an:nN}\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 (an)(a_n) is given as decreasing, meaning that an+1ana_{n+1} \leq a_n for all nn.
    • Additionally, it is bounded below, which means there exists a number mm such that anma_n \geq m for all nn.
  2. Monotone Convergence Theorem (for real sequences):

    • The monotone convergence theorem tells us that every monotone sequence that is bounded converges.
    • Since (an)(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: limnan=inf{an:nN}.\lim_{n \to \infty} a_n = \inf \{a_n : n \in \mathbb{N}\}.

Final Conclusion:

The sequence (an)(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.

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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