. Prove that an→L = Jan] → |L]. (Make cas
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Transcribed Image Text:### Problem 5: Limit Proof Problem
**Objective**: Demonstrate the following mathematical proof:
**Statement**: Prove that if \( a_n \to L \), then \( |a_n| \to |L| \).
**Instructions**: Make cases and use Theorem 5.3B for the proof.
**Key Concepts**:
- The notation \( a_n \to L \) indicates that the sequence \( a_n \) converges to the limit \( L \).
- Similarly, \( |a_n| \to |L| \) means that the sequence of absolute values \( |a_n| \) converges to the absolute value of the limit \( L \).
### Explanation
To successfully prove this statement, consider the different possible cases for the limit \( L \) and approach the proof accordingly. Applying Theorem 5.3B will aid in structuring the proof based on known mathematical properties and theorems.
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Given that an is a sequence which converges to L that is,
.
Then we have to prove that |an| converges to |L | that is,
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