Chapter 7: Q. 24 (page 592)
Let \(\left \{ a_k \right \}\) be the sequence \(a_1=3, a_2=3.1, a_3=3.14, a_4=3.141, ...\) 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.
Short Answer
Part a. For each \(k\), the number \(a_{k}\) has a finite decimal representation. Hence, \(a_k\) is a rational number for each positive integer \(k\).
Part b. The sequence is increasing because each \(a_{k+1}\) is obtained from \(a_k\) by adding one more significant place after the decimal.
Part c. An upper bound for the sequence \(\left \{ a_k \right \}\) is \(3.2\)
Part d. The least upper bound of the sequence \(\left \{ a_k \right \}\) is \(\pi\).
Part e. The Least Upper Bound Axiom does not apply to the set of rational numbers because the sequence \(\left \{ a_1,a_2,a_3,a_4,... \right \}\) is set of rational numbers bounded above, but its least upper bound is \(\pi\) which is not rational.