Proof Prove Theorem 9.5 for a nonincreasing sequence.
To Prove: If the sequence is bounded and monotonic for a non-increasing sequence then it is convergent.
Let the sequence be non-increasing. To keep things simple, assume that each term in the sequence is positive.
Since the sequence is bounded, there exists a lower bound.
For the complete axiom, there is a greatest lower bound.
Now, it follows that and therefore, cannot be a lower bound for the sequence.
Consequently, at least one term of is less than i
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