Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
General Math
Calculus
Differential Equations
Topology and Analysis
Linear and Abstract Algebra
Differential Geometry
Set Theory, Logic, Probability, Statistics
MATLAB, Maple, Mathematica, LaTeX
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Mathematics
Topology and Analysis
Connectedness and Intervals in R .... Stromberg, Theorem 3.47 .... ....
Reply to thread
Message
[QUOTE="Math Amateur, post: 6776107, member: 203675"] I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.47 on page 107 ... ... Theorem 3.47 and its proof read as follows:[ATTACH=CONFIG]9153._xfImport[/ATTACH] In the second paragraph of the above proof by Stromberg we read the following: " ... ... Since [MATH]U[/MATH] is open we can choose [MATH]c' \gt c[/MATH] such that [MATH][ c, c' ] \subset U \cap [a, b][/MATH] ... ... " My question is as follows: Can someone please demonstrate rigorously why/how ... [MATH]U[/MATH] is open [MATH]\Longrightarrow[/MATH] we can choose [MATH]c' \gt c[/MATH] such that [MATH][ c, c' ] \subset U \cap [a, b] [/MATH] ... ... Indeed I can see that ... [MATH]U[/MATH] is open [MATH]\Longrightarrow \exists[/MATH] an open ball [MATH]B_r(c) = \ ] c - r, c + r [ \ \subset U[/MATH] ... ...but how do we conclude from here that [MATH]U[/MATH] is open [MATH]\Longrightarrow[/MATH] we can choose [MATH]c' \gt c[/MATH] such that [MATH][ c, c' ] \subset U \cap [a, b][/MATH] ... ...*** EDIT *** It may be that the solution is to choose [MATH]s \lt r[/MATH] so that [MATH][ c, c + s] \subset U[/MATH] where [MATH]c' = c + s[/MATH] ... but how do we ensure this interval also belongs to [MATH][a, b][/MATH] ... ... ? Help will be appreciated ... ... Peter =======================================================================================Stromberg uses slightly unusual notation for open intervals in [MATH]\mathbb{R}[/MATH] and [MATH]\mathbb{R}^{\#} = \mathbb{R} \cup \{ \infty , - \infty \}[/MATH] so I am providing access to Stromberg's definition of intervals in [MATH]\mathbb{R}^{ \#} [/MATH] ... as follows: [ATTACH=CONFIG]9152._xfImport[/ATTACH] Hope that helps ... Peter [/QUOTE]
Insert quotes…
Post reply
Forums
Mathematics
Topology and Analysis
Connectedness and Intervals in R .... Stromberg, Theorem 3.47 .... ....
Back
Top