Determine whether each of the statements that follow is true...

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample. (a) True or False: Every sequence is a function. (b) True or False: The third term of the sequence $\{k+1\}_{k=1}^{\infty}$ is $4 .$ (c) True or False: The third term of the sequence $\left\{k^{2}\right\}_{k=2}^{\infty}$ is $9 .$ (d) True or False: Every sequence of real numbers is either increasing or decreasing. (e) True or False: Every sequence of numbers has a smallest term. (f) True or False: Every recursively defined sequence has an infinite number of distinct outputs. (g) True or False: Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound. (h) True or False: Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: Every sequence is a function.
(b) True or False: The third term of the sequence $\{k+1\}_{k=1}^{\infty}$ is $4 .$
(c) True or False: The third term of the sequence $\left\{k^{2}\right\}_{k=2}^{\infty}$ is $9 .$
(d) True or False: Every sequence of real numbers is either increasing or decreasing.
(e) True or False: Every sequence of numbers has a smallest term.
(f) True or False: Every recursively defined sequence has an infinite number of distinct outputs.
(g) True or False: Every sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.
(h) True or False: Every monotonic sequence has an upper bound, a lower bound, or both an upper bound and a lower bound.

Key Concepts

Minimum Element in a Sequence

The minimum element of a sequence is a term that is smaller than or equal to every other term in the sequence. While some sequences may have a smallest term, especially when considering finite segments or sequences with a clearly defined minimum due to their structure, not all sequences possess such a minimum within their domain.

Recursive Sequence Definitions

Recursively defined sequences are specified by stating one or more initial terms and providing a rule that relates each term to the previous term(s). Recognizing that recursion may sometimes lead to repeated values is important when considering properties such as the distinctness of outputs, rather than assuming perpetual generation of new values.

Boundedness of Sequences

Boundedness involves whether a sequence is confined within an upper bound, a lower bound, or both. A sequence is said to be bounded if there exists a real number that serves as an upper limit or lower limit for all its terms. This concept is key in analyzing convergence and the overall behavior of sequences.

Term Evaluation and Indexing

The evaluation of a sequence’s individual terms is based on its formula and the indexing of its domain. This concept involves substituting specific indices into the formula defining the sequence to determine its corresponding outputs, and understanding the starting index is crucial to correctly interpreting the sequence.

Sequence as a Function

A sequence is fundamentally a function whose domain is a subset of the integers (often the natural numbers) and whose codomain is a set such as the real numbers. Understanding that a sequence is a function allows us to apply function-based reasoning (e.g., domain, range, and rules of assignment) to the study of sequences.

Monotonicity of Sequences

Monotonicity refers to a sequence consistently increasing (or non-decreasing) or decreasing (or non-increasing) with each successive term. A clear grasp of the definitions of increasing, decreasing, and non-monotonic sequences is essential in assessing properties related to their behavior over the domain.

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