Absolute Value of Limits Proof

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SUMMARY

The discussion centers on proving that if the sequence \( b_n \) converges to \( b \), then the sequence of absolute values \( |b_n| \) converges to \( |b| \). Participants emphasize the importance of clearly stating assumptions and the desired conclusion to simplify the proof process. A counterexample is provided to illustrate that the converse does not hold: while \( |(-1)^n| \) converges to 1, the sequence \( (-1)^n \) does not converge. This highlights the necessity of understanding the properties of limits in relation to absolute values.

PREREQUISITES
  • Understanding of limit definitions in calculus
  • Familiarity with sequences and convergence
  • Knowledge of absolute value properties
  • Experience with counterexamples in mathematical proofs
NEXT STEPS
  • Study the formal definition of limits in calculus
  • Explore properties of absolute values in sequences
  • Learn about convergence criteria for sequences
  • Investigate additional counterexamples related to limits and absolute values
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Students in calculus, mathematicians focusing on real analysis, and anyone interested in understanding the properties of limits and sequences.

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Homework Statement



Show that if bn→b, then the sequence of absolute values |bn| converges to |b|.

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The Attempt at a Solution



I've been proving various properties of limits, including product of limits and sum of products, but have been having trouble making progress with the approach to absolute value of limits. I was also wondering is the converse of this is true, that is if |bn|→|b|, is it also true that (bn)→b
 
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Not at all, for example |(-1)^n| -> 1, but (-1)^n doesn't converge at all!
How about you start by writing down formally the assumption and what you want to prove. You will see that it's pretty straight forward then.
 

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