Let
{
𝑎
𝑛
}
𝑛
=
0
∞
be a sequence.

Which of the following statements is (equivalent to) the definition of the concept that "the sequence is not increasing"?

Select all the correct answers.

Question

Let 
{
𝑎
𝑛
}
𝑛
=
0
∞
 be a sequence.

Which of the following statements is (equivalent to) the definition of the concept that "the sequence is not increasing"?

Select all the correct answers.

BlackTom AI · Accepted Answer

The question asks which statements are equivalent to the definition of the sequence not increasing. We will examine each option in turn and explain why it may or may not capture the intended meaning.

Option 1 (from answer_options): ∃ n, m ∈ N, s.t. n < m and a_n ≥ a_m
This option asserts that there exists at least one pair of indices with n < m where the earlier term is greater than or equal to the later term. This exactly captures the idea that the sequence is not strictly increasing: if such a pair exists, the sequence cannot be strictly increasing because, for a strictly increasing sequence, every earlier term would be strictly less than every later term. Therefore, this condition is a correct characterization of "not increasing" in the usual sense. In other words, the existence of a backward or non-increasing step suffices to violate strict increase.

Option 2 (from answer_options): ∀ n, m ∈ N, n < m ⇒ a_n ≥ a_m
This statement says that for every pair with n < m, we have a_n ≥ a_m. That would mean the sequence is nonincreasing (or monotone decreasing in the broadest sense): every earlier term is at least as large as every later term. While such a property certainly implies the sequence is not increasing, it is stronger than the usual definition of "not increasing" (which only requires that at least one pair violates strict increase). Therefore, as an equivalence to "the sequence is not increasing", this universal condition is too strong and is not equivalent in general. It rules out sequences that sometimes go up but are not strictly increasing overall. However, if one adopted the interpretation that "not increasing" means the sequence is nonincreasing for all pairs, then this would be aligned. In standard mathematics, though, the existential condition in Option 1 is the standard equivalent.

Option 3 (from answer_options): ∃ n ∈ N, s.t. a_n ≥ a_n + 2
This is an impossible inequality because a_n ≥ a_n + 2 cannot hold for any real (or most common) sequences. It asserts something false for all n, so it cannot be equivalent to any reasonable definition of not increasing. This option is incorrect because it is universally false and does not relate to the behavior of the sequence across indices.

Option 4 (from answer_options): ∃ n, m ∈ N, s.t. n < m and a_n ≥ a_m
This is identical in meaning to Option 1 (duplicate wording in a different line). Since it expresses the same existential condition about existence of a pair with n < m and a_n ≥ a_m, it serves as another correct formulation of not being strictly increasing. Thus, it is effectively correct for the same reason as Option 1.

Option 5 (from answer_options): ∀ n ∈ N, a_n ≥ a_n + 1
This inequality is also never true for any real sequence, since a_n cannot be simultaneously equal to or greater than a_n + 1. It is logically impossible, so it does not describe a valid property of sequences and has nothing to do with the concept of not increasing. Therefore, this option is incorrect.

In summary, the meaningful equivalence to not increasing hinges on the existence of a single counterexample to strict increase (Option 1). Option 2 describes a stronger, universal nonincreasing behavior and is not equivalent in general. Options 3 and 5 are invalid constructions that cannot characterize not increasing, and Option 4 repeats the correct existential condition from Option 1.

MAT137Y1 LEC 20249: Calculus with Proofs (all lecture sections) Pre-Class Quiz 51 (11.3 and 11.4)

Let { 𝑎 𝑛 } 𝑛 = 0 ∞ be a sequence. Which of the following statements is (equivalent to) the definition of the concept that "the sequence is not increasing"? Select all the correct answers.

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