Proving the Non-Negativity of a Decreasing Sequence with Limit 0

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SUMMARY

The discussion focuses on proving the non-negativity of a decreasing sequence that converges to 0. The sequence, denoted as \( a_n \), must satisfy two conditions: it is decreasing and its limit approaches 0. The proof requires demonstrating that for any natural number \( n \), \( a_n \geq 0 \). A critical aspect of the proof involves using the limit definition, specifically that if \( a_{\bar{n}} < 0 \), a suitable choice of epsilon would invalidate the limit condition.

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Analysis Proof Help!

If anyone could give me a hint on how to start this I would appreciate it! I am struggling with proofs! Thanks!

Given a sequence (asub(n))
s.t.
(i) the sequence is decreasing
(ii) the limit of the sequence is 0.
Prove rigorously that an is greater than or equal to 0 for every n contained in the natural numbers
 
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[itex]a_n\rightarrow 0[/itex] if [itex]\forall \epsilon >0\exists \bar{n}\backepsilon' \forall n>\bar{n}, |a_n|<\epsilon[/itex].
Try to prove that, if [itex]a_{\bar{n}}<0[/itex], with a particular choice of epsilon the limit definition is not verified.
Remember to consider the fact that the sequence is decreasing
 

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