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zachsdado
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Homework Statement
prove that a monotone sequence which has a bounded subsequence is bounded
A sequence is considered bounded if all of its values fall within a certain range. This means that there is a maximum and minimum value for the sequence, and all values in between are also included.
To prove that a sequence is bounded, you can either show that the sequence is increasing and has an upper bound, or that the sequence is decreasing and has a lower bound. Another way to prove boundedness is by using the squeeze theorem, which compares the sequence to another sequence with known bounds.
Yes, a sequence can still be bounded even if it has an infinite number of terms. As long as all the terms fall within a certain range, the sequence is considered bounded.
No, a sequence cannot be both increasing and decreasing. An increasing sequence has values that are getting larger, while a decreasing sequence has values that are getting smaller. Therefore, a sequence can only have one of these properties at a time.
No, a bounded sequence does not always converge. A sequence can be bounded but still have values that fluctuate and do not approach a certain value. Convergence also depends on the behavior of the terms in the sequence, not just the bounds.