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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 some remarks by Garling made after the proof of Theorem 3.1.1 ...Garling's statement and proof of Theorem 3.1.1 (together with the interesting remarks) reads as follows:

View attachment 9031

View attachment 9032

My questions on the remarks after the proof are as follows:

In the remarks after the proof of Theorem 3.1.1 we read the following:

" ... ... There exists a least positive integer, \(\displaystyle q_0\), say, such that 1\(\displaystyle /q_0 \lt y - x\) ... ... "Can someone please explain exactly why this is true ... ..

In the remarks after the proof of Theorem 3.1.1 we read the following:

" ... ... and there then exists a least integer, \(\displaystyle p_0\), say, such that \(\displaystyle x \lt p_0 / q_0\) ... ..."

In the remarks after the proof of Theorem 3.1.1 we read the following:

" ... ... Then \(\displaystyle x \lt p_0 / q_0 \lt y\) and \(\displaystyle r_0 = p_o / q_0\) is uniquely determined ... ... "Can someone please explain exactly why this is true ... ..Help will be appreciated ...

Peter

I am focused on Chapter 3: Convergent Sequences

I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1 ...Garling's statement and proof of Theorem 3.1.1 (together with the interesting remarks) reads as follows:

View attachment 9031

View attachment 9032

My questions on the remarks after the proof are as follows:

**Question 1**In the remarks after the proof of Theorem 3.1.1 we read the following:

" ... ... There exists a least positive integer, \(\displaystyle q_0\), say, such that 1\(\displaystyle /q_0 \lt y - x\) ... ... "Can someone please explain exactly why this is true ... ..

**Question 2**In the remarks after the proof of Theorem 3.1.1 we read the following:

" ... ... and there then exists a least integer, \(\displaystyle p_0\), say, such that \(\displaystyle x \lt p_0 / q_0\) ... ..."

**Question 3**

In the remarks after the proof of Theorem 3.1.1 we read the following:

" ... ... Then \(\displaystyle x \lt p_0 / q_0 \lt y\) and \(\displaystyle r_0 = p_o / q_0\) is uniquely determined ... ... "Can someone please explain exactly why this is true ... ..Help will be appreciated ...

Peter

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