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Suppose you know that $ \left\{ a_n \right\} $ is a decreasing sequence and all its terms lie between the numbers 5 and 8. Explain why the sequence has a limit. What can you say about the value of the limit?

$\left\{a_{n}\right\}$ is decreasing, so $5 \leq L<8$

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

for this problem. You're given that hey and is decreasing, and here you see that it's between five and eight for each end. This means that the sequence is bounded. You have an upper and lower bound. That means it's bounded. Now, why must this sequence have a limit? Let's answer that using a the room since A M is bounded and monotone by the monotone sequence, their own and converges. On the other hand, we see that the sequence is bounded between five and eight, so it's not possible for the limit to go outside of the bounds. So we can say that the limit is in between five and eight does not necessarily have to be fiber eight again five and eight or bounds. It's not possible for the sequence escaped the bounds. Therefore, the limit must stay inside that lower upper bounds. So the limits between five and eight that's our final answer