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MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 52 (11.5 and 11.6)

Multiple choice

Let  { 𝑎 𝑛 } 𝑛 = 0 ∞  be a POSITIVE sequence.  Which of the following statements must be true? Select all the correct answers. 

Options
A.IF the sequence is bounded, THEN it is eventually monotonic.
B.IF the sequence is convergent, THEN it is bounded.
C.IF the sequence is bounded above, THEN it is convergent.
D.IF the sequence is non-decreasing, THEN it is convergent.
E.IF the sequence is non-increasing, THEN it is convergent.
F.IF the sequence is not bounded, THEN it is divergent.
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Step-by-Step Analysis
Before evaluating each option, recall the given conditions: a_n is a positive sequence, so a_n > 0 for all n. From this, we have a basic lower bound 0 for all terms, which can be useful when considering convergence and monotonicity. Option 1: 'IF the sequence is bounded, THEN it is eventually monotonic.' This statement is not necessarily true. A sequence can be bounded yet fail to become monotone eventually; boundedness does not imply eventual monotonicity. For instance......Login to view full explanation

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