The discussion revolves around proving a property of convergent sequences, specifically addressing the condition where the limit of a sequence is less than a certain value. Participants explore the implications of this condition and how it relates to the behavior of the sequence.
The discussion is active, with participants offering insights and corrections. Some have provided potential approaches to the proof, while others have raised questions about the assumptions made in the original post. There is a mix of interpretations being explored, and guidance has been offered regarding the definition of convergence.
Participants note the importance of correctly interpreting the conditions of the problem, particularly the relationship between the limit and the value p. There is an acknowledgment of the need for clarity in the definitions being used.
Counterexample: Let an = 1 - (1/(n+1)). Then [itex]\lim_{n\rightarrow\infty}a_n = 1[/itex]. Choose p = 1/10 < 1. Then [itex]\forall N \in \mathbb{N} \exists n\ge N : a_n > p[/itex].kreil said:
Not necessarily. Let an = 1/(n+1). Then [itex]\lim_{n\rightarrow\infty} a_n = 0[/itex]. If you choose p=9/10 > 0, all the an's near l are less than p. I think a qualifier is missing somewhere.Physics Monkey said:
That's probably the right question, then. In that case, kreil should appeal to the definition of convergence.kreil said: