Proof Prove Theorem 9.5 for a nonincreasing sequence.
To Prove: Bounded monotonic sequence theorem for non-increasing sequence.
If a sequence is bounded monotonic and non-increasing, then it converges
Assume that the sequence is non-increasing. To keep things simple, also make the assumption that each term in the sequence is positive.
As the sequence is bounded, there exists a lower bound
For the completeness axiom, there is a greatest lower bound.
Now, it follows that and therefore, cannot be a lower bound for the sequence
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