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Chapter 1: Problem 10

If \(a, b \in \mathbb{R}\) with \(a

Short Answer

Expert verified
In summary, by constructing two sequences, one of rational numbers (\(r_n = a + \frac{b-a}{2n}\)) and one of irrational numbers (\(I_n = r_n + \frac{\sqrt{2}}{2n}\)), we showed that there exist infinitely many rational and irrational numbers between two real numbers \(a\) and \(b\) (\(a<b\)). Both sequences satisfy the conditions \(a<r_n<b\) and \(a<I_n<b\), respectively, for all positive integers \(n\).

Step by step solution

01

Construct a sequence of rational numbers between \(a\) and \(b\)

To construct a sequence of rational numbers between \(a\) and \(b\), let's consider the midpoint \(\frac{a+b}{2}\). This midpoint is also a real number, and since \(a<b\), it falls between \(a\) and \(b\). Now, let's generate a sequence of rational numbers \(r_n\) as follows: \(r_n = a + \frac{b-a}{2n}\), where \(n\) is a positive integer. The sequence of \(r_n\) are rational numbers because the sum and difference of rational numbers are always rational. Since \(n\) is a positive integer, we have \(0 < \frac{b-a}{2n} < \frac{b-a}{2}\), which implies \(a < r_n < b\). Therefore, we can construct infinitely many rational numbers between \(a\) and \(b\) by varying the value of \(n\).
02

Construct a sequence of irrational numbers between \(a\) and \(b\)

To construct a sequence of irrational numbers between \(a\) and \(b\), let's consider adding an irrational number to the sequence of rational numbers \(r_n\) we constructed earlier. Let \(I_n\) be the sequence of irrational numbers, where: \(I_n = r_n + \frac{\sqrt{2}}{2n}\), where \(n\) is a positive integer. Since \(r_n\) is rational and \(\frac{\sqrt{2}}{2n}\) is irrational, their sum \(I_n\) is also irrational. Now we need to show that \(I_n\) is indeed between \(a\) and \(b\). We know that \(a<r_n\) and \(r_n<I_n\), which implies \(a<I_n\). We also have \(I_n<r_n+\frac{b-a}{2}\), which implies \(I_n<b\). Therefore, we can construct infinitely many irrational numbers between \(a\) and \(b\) by varying the value of \(n\). In conclusion, we have shown that there exist infinitely many rational numbers as well as infinitely many irrational numbers between \(a\) and \(b\), as required.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Numbers
Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. In mathematical terms, a rational number is any number that can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q eq 0 \).

Rational numbers are integral to the field of real analysis because they help us understand the density properties of the real number line. In the exercise above, we showed that there are infinitely many rational numbers between any two distinct real numbers \( a \) and \( b \). To construct these numbers, we created a sequence \( r_n = a + \frac{b-a}{2n} \), which uses the properties of rational numbers to ensure each \( r_n \) is also a rational number.

  • The sum and difference of two rational numbers is rational.
  • Multiplying a rational number by an integer results in a rational number.
  • Between any two rational numbers, there are infinitely many other rational numbers.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. These numbers have non-terminating and non-repeating decimal expansions. Examples include \( \sqrt{2} \), \( \pi \), and \( \mathrm{e} \).

In terms of the exercise, to find infinitely many irrational numbers between two real numbers \( a \) and \( b \), we utilized a previously established sequence of rational numbers \( r_n \) and added an irrational component to it. This created a new sequence, \( I_n = r_n + \frac{\sqrt{2}}{2n} \). The term \( \frac{\sqrt{2}}{2n} \) is necessarily irrational due to \( \sqrt{2} \) being irrational, and hence, the sum \( I_n \) remains irrational.

  • Irrationals fill gaps in the real number line not covered by rational numbers.
  • The sum of a rational number and an irrational number is always irrational.
  • There are uncountably many irrational numbers, as opposed to countably many rationals.
Sequences
Sequences play a critical role in real analysis. A sequence is an ordered list of numbers defined by some rule or formula. In our exercise, sequences helped exhibit the infinite presence of rational and irrational numbers between two given real numbers \( a \) and \( b \).

Using sequences allows us to systematically and infinitely generate numbers that adhere to specific properties. For example, the rational sequence \( r_n = a + \frac{b-a}{2n} \) founded on the arithmetic progression. Additionally, the irrational sequence \( I_n = r_n + \frac{\sqrt{2}}{2n} \) demonstrates how slight perturbations involving irrational numbers always yield irrationality.

  • Sequences form the backbone for more advanced mathematical concepts like limits.
  • They allow for constructing examples in a step-by-step manner.
  • Convergence of sequences is a fundamental notion in understanding functions and continuity.

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Most popular questions from this chapter

Show that \(n ! \leq 2^{-n}(n+1)^{n}\) for every \(n \in \mathbb{N}\), and that equality holds if and only if \(n=1\).

If \(A\) and \(B\) are any countable sets, then show that the set \(A \times B:=\\{(a, b):\) \(a \in A\) and \(b \in B\\}\) is also countable. (Hint: Write \(A \times B=\cup_{b \in B} A \times\\{b\\} .\) )

Let \(f:[a, b] \rightarrow \mathbb{R}\) be a function and let \(c\) be a point of \((a, b)\). (i) If \(f\) is monotonically (resp. strictly) increasing on \([a, c]\) and on \([c, b]\), then show that \(f\) is monotonically (resp. strictly) increasing on \([a, b]\). (ii) If \(f\) is convex (resp. strictly convex) on \([a, c]\) and on \([c, b]\), then is it true that \(f\) is convex (resp. strictly convex) on \([a, b] ?\)

Let \(S\) be a nonempty subset of \(\mathbb{R} .\) If \(S\) is bounded above, then show that the set \(U_{S}=\\{\alpha \in \mathbb{R}: \alpha\) is an upper bound of \(S\\}\) is bounded below, \(\min U_{S}\) exists, and sup \(S=\min U_{S}\). Likewise, if \(S\) is bounded below, then show that the set \(L_{S}=\\{\beta \in \mathbb{R}: \beta\) is a lower bound of \(S\\}\) is bounded above, \(\max L_{S}\) exists, and inf \(S=\max L_{S}\).

Give an example of \(f:(0,1) \rightarrow \mathbb{R}\) such that \(f\) is (i) strictly increasing and convex, (ii) strictly increasing and concave, (iii) strictly decreasing and convex, (iv) strictly decreasing and concave.
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