Let {ak} be the sequence a1=3, a2=3.1, a3=3.14, a4=3.141, …That is, each term ak contains the fi

step 1 For this problem, we are told to let AK be the sequence A1 equals 3, A2 equals 3 .1, A3 equals 3

Let {ak} be the sequence a1=3, a2=3.1, a3=3.14, a4=3.141,...

Let $\left\{a_{k}\right\}$ be the sequence $a_{1}=3, a_{2}=3.1, a_{3}=3.14$, $a_{4}=3.141, \ldots$ That is, each term $a_{k}$ contains the first $k$ decimal digits of $\pi$. (a) Explain why $a_{k}$ is a rational number for each positive integer $k$ (b) Explain why the sequence $\left\{a_{k}\right\}$ is increasing. (c) Provide an upper bound for the sequence $\left\{a_{k}\right\}$. (d) What is the least upper bound of the sequence $\left\{a_{k}\right\}$ ? (e) Use this sequence to explain why the Least Upper Bound Axiom does not apply to the set of rational numbers.

Let $\left\{a_{k}\right\}$ be the sequence $a_{1}=3, a_{2}=3.1, a_{3}=3.14$, $a_{4}=3.141, \ldots$ That is, each term $a_{k}$ contains the first $k$ decimal digits of $\pi$.
(a) Explain why $a_{k}$ is a rational number for each positive integer $k$
(b) Explain why the sequence $\left\{a_{k}\right\}$ is increasing.
(c) Provide an upper bound for the sequence $\left\{a_{k}\right\}$.
(d) What is the least upper bound of the sequence $\left\{a_{k}\right\}$ ?
(e) Use this sequence to explain why the Least Upper Bound Axiom does not apply to the set of rational numbers.

Key Concepts

Least Upper Bound (Supremum)

The least upper bound or supremum of a sequence is the smallest number that is no less than any term of the sequence. In this case, while ? is not a term in the sequence, it is the result that the sequence approaches as more digits of ? are included, making ? the least upper bound of the sequence.

Completeness Axiom

The Completeness Axiom states that every nonempty set of real numbers that is bounded above has a least upper bound in the real numbers. The sequence provided comprises rational numbers that approach the irrational number ?. Since ? is not rational, the set of rational numbers lacks a least upper bound in the rationals, illustrating that the Completeness Axiom does not hold in the rational number system.

Upper Bound

An upper bound of a set or sequence is a number that is greater than or equal to every term in the set. For this sequence, ? serves as a natural upper bound because none of the terms exceed ?. Providing an upper bound is important for analyzing the behavior of sequences and assessing their limits.

Monotonic (Increasing) Sequence

A sequence is said to be increasing or monotonic if every term is greater than or equal to the previous term. In the given sequence, each subsequent term appends an additional digit from ?, resulting in a number that is a closer approximation to ?. Since these terms add more precision in a way that increases the value while remaining below ?, the sequence is increasing.

Rational Numbers

A rational number is any number that can be expressed as the quotient of two integers. In the context of the sequence, each term is a terminating decimal, and any terminating decimal can be written as a fraction where the denominator is a power of 10. This property ensures that every term in the sequence is rational.

Transcript

Please give Ace some feedback