Understanding Garling's Corollary 3.2.7 on Real Numbers and Rational Sequences

In summary, Garling's statement and proof of Corollary 3.2.7 (together with Proposition 3.2.6 which is mentioned in Corollary 3.2.7 ... ) reads as follows:There is a strictly increasing rational sequence $x-1\ <\ r_1\ <\ r_2\ <\ r_3\ <\ \cdots\ <\ x$ that converges to $x$.
  • #1
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I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...

I am focused on Chapter 3: Convergent Sequences

I need some help to fully understand the proof of Corollary 3.2.7 ...Garling's statement and proof of Corollary 3.2.7 (together with Proposition 3.2.6 which is mentioned in Corollary 3.2.7 ... ) reads as follows:

View attachment 9037
My questions related to the above Corollary are as follows:
Question 1

In the above proof of Corollary 3.2.7 we read the following:

" ... ... Arguing recursively, let \(\displaystyle r_n\) be the 'best' rational with \(\displaystyle \text{max} ( x - \frac{1}{n} , r_{ n - 1 } ) \lt r_n \lt x\) ... ... Can someone please explain (preferably in some detail) what is going on here ... how do we arrive at the expression

\(\displaystyle \text{max} ( x - \frac{1}{n} , r_{ n - 1 } ) \lt r_n \lt x\) ... ... "Question 2

In the above proof of Corollary 3.2.7 we read the following:

" ... ... let \(\displaystyle s_n\) be the 'best' rational with

\(\displaystyle x \lt s_n \lt \text{min} ( x + \frac{1}{n}, s_{ n - 1 } )\) ... ... "Can someone please explain (preferably in some detail) what is going on here ... how do we arrive at the expression

\(\displaystyle x \lt s_n \lt \text{min} ( x + \frac{1}{n}, s_{ n - 1 } )\) ... ... ?
Help will be appreciated ...

Peter==========================================================================================The post above mentions Theorem 3.1.1 and alludes to the remarks made after the proof of Theorem 3.1.1 ... so I am providing text of the theorem and the relevant remarks ... as follows:View attachment 9038
View attachment 9039

Hope that helps ...

Peter
 

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  • #2
Peter said:
Question 1

In the above proof of Corollary 3.2.7 we read the following:

" ... ... Arguing recursively, let \(\displaystyle r_n\) be the 'best' rational with \(\displaystyle \text{max} ( x - \frac{1}{n} , r_{ n - 1 } ) \lt r_n \lt x\) ... ... Can someone please explain (preferably in some detail) what is going on here ... how do we arrive at the expression

\(\displaystyle \text{max} ( x - \frac{1}{n} , r_{ n - 1 } ) \lt r_n \lt x\) ... ... "

Garling is just trying to construct a sequence of rational numbers that is strictly increasing and converges to $x$. If $x$ itself is rational, we can just take the sequence
$$\left(x-\frac1n\right)_{n=1}^\infty$$
but $x$ might be irrational whereas we want a rational sequence. By choosing
$$\max\left(x-\frac1n,r_{n-1}\ <\ r_n\right)$$
we are ensuring that the sequence is rational and strictly increasing, and approaches arbitrarily close to $x$. Theorem 3.1.1 guarantees the existence of a strictly increasing rational sequence
$$x-1\ <\ r_1\ <\ r_2\ <\ r_3\ <\ \cdots\ <\ x$$
but we also want the $r_n$ to converge to $x$. This is done by ensuring each $r_n$ is at least $x-\dfrac1{n+1}$.

The other question is similar.
 
  • #3
Olinguito said:
Garling is just trying to construct a sequence of rational numbers that is strictly increasing and converges to $x$. If $x$ itself is rational, we can just take the sequence
$$\left(x-\frac1n\right)_{n=1}^\infty$$
but $x$ might be irrational whereas we want a rational sequence. By choosing
$$\max\left(x-\frac1n,r_{n-1}\ <\ r_n\right)$$
we are ensuring that the sequence is rational and strictly increasing, and approaches arbitrarily close to $x$. Theorem 3.1.1 guarantees the existence of a strictly increasing rational sequence
$$x-1\ <\ r_1\ <\ r_2\ <\ r_3\ <\ \cdots\ <\ x$$
but we also want the $r_n$ to converge to $x$. This is done by ensuring each $r_n$ is at least $x-\dfrac1{n+1}$.

The other question is similar.

Thanks for that helpful post ,,,

The idea of the proof is clear to me now ...

Peter
 

FAQ: Understanding Garling's Corollary 3.2.7 on Real Numbers and Rational Sequences

1. What is Garling's Corollary 3.2.7?

Garling's Corollary 3.2.7 is a mathematical theorem that states that every real number is the limit of a rational sequence. In other words, for any real number, there exists a sequence of rational numbers that converges to that real number.

2. Why is Garling's Corollary 3.2.7 important?

This corollary is important because it helps us understand the relationship between real numbers and rational numbers. It shows that even though real numbers may seem more complex, they can still be represented by a sequence of simpler rational numbers.

3. How is Garling's Corollary 3.2.7 related to the concept of convergence?

Garling's Corollary 3.2.7 is directly related to the concept of convergence. It states that a sequence of rational numbers converges to a real number, meaning that as the terms in the sequence get closer and closer to the real number, the sequence "converges" to that real number.

4. Can you provide an example of Garling's Corollary 3.2.7 in action?

Sure, let's take the real number pi (π) as an example. We know that pi is an irrational number, meaning it cannot be expressed as a fraction of two integers. However, using Garling's Corollary 3.2.7, we can find a sequence of rational numbers that converges to pi. One such sequence is 3, 3.1, 3.14, 3.141, 3.1415, ... As we continue this sequence, the terms get closer and closer to pi, thus satisfying the corollary.

5. How does Garling's Corollary 3.2.7 relate to the decimal representation of real numbers?

Garling's Corollary 3.2.7 is closely related to the decimal representation of real numbers. It shows that any real number can be approximated by a sequence of rational numbers, which is essentially what the decimal representation of a real number is. This corollary helps us understand that even though we may not be able to write out the exact decimal representation of a real number, it can still be approximated by a sequence of rational numbers.

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