- #1
Math Amateur
Gold Member
MHB
- 3,990
- 48
- TL;DR Summary
- For any given real number x, there exists a strictly increasing sequence of rationals that converges to x as a limit ... same for a strictly decreasing sequence of rationals ...
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:
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 ##r_n## be the 'best' rational with##\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
##\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 ##s_n## be the 'best' rational with
##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
##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:
Hope that helps ...
Peter
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:
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 ##r_n## be the 'best' rational with##\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
##\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 ##s_n## be the 'best' rational with
##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
##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:
Peter