- #1
anhtudo
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- TL;DR Summary
- What I don't understand is how he proves that G = N.
I don't think it is logical to let b = n as it can not be derived from the definition of G that b is in G.
Thanks.
Lemma 1.2.3 - Ethan.D.Bloch - The Real Numbers and Real Analysis
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- Thread starter anhtudo
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In summary, Lemma 1.2.3 is a mathematical statement found in Ethan.D.Bloch's book, "The Real Numbers and Real Analysis". It serves as a stepping stone to prove larger theorems and propositions in the field of real analysis. Ethan.D.Bloch is a mathematician and professor at the University of Alabama, specializing in real analysis, topology, and number theory. The real numbers are a set of numbers that include all rational and irrational numbers and are used to measure quantities in the physical world. Real analysis differs from other branches of mathematics in its focus on continuous and infinite objects, and Lemma 1.2.3 plays a crucial role in establishing the validity of theorems and solving complex problems in
- #2
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It is written a bit confusing, but correct. Forget ##b##.
We have ##n\in G## and ##p=s(n)##. This implies ##p \in G## because there is some ##n \in \mathbb{N}## such that ##s(n)=p##. Hence ##s(n) \in G##. Therefore we have ##1\in G## and all successors of elements of ##G## are in ##G##, too, i.e. ##\mathbb{N} \subseteq G##.
We have ##n\in G## and ##p=s(n)##. This implies ##p \in G## because there is some ##n \in \mathbb{N}## such that ##s(n)=p##. Hence ##s(n) \in G##. Therefore we have ##1\in G## and all successors of elements of ##G## are in ##G##, too, i.e. ##\mathbb{N} \subseteq G##.
- #3
anhtudo
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Thank you.
1. What is Lemma 1.2.3 in Ethan.D.Bloch's "The Real Numbers and Real Analysis"?
Lemma 1.2.3 is a mathematical statement that is used to prove a theorem in the book "The Real Numbers and Real Analysis" by Ethan.D.Bloch. It is a specific step or subproof that is essential in proving the main theorem.
2. Who is Ethan.D.Bloch and why is his book important?
Ethan.D.Bloch is a mathematician and author who has written several books on real analysis and other mathematical topics. His book "The Real Numbers and Real Analysis" is important because it provides a comprehensive and rigorous introduction to the concepts of real numbers and real analysis, which are essential for understanding advanced mathematics.
3. What are real numbers and why are they important in mathematics?
Real numbers are numbers that can be represented on a number line and include both rational and irrational numbers. They are important in mathematics because they are used to describe quantities and measurements in the real world and are the foundation for more complex mathematical concepts and theories.
4. How is Lemma 1.2.3 used in real analysis?
Lemma 1.2.3 is used in real analysis as a crucial step in proving a theorem. It often involves breaking down a complex problem into smaller, more manageable parts and using mathematical techniques and definitions to show that each part is true, leading to the proof of the main theorem.
5. Can Lemma 1.2.3 be applied to other mathematical fields?
Yes, Lemma 1.2.3 can be applied to other mathematical fields as it is a general proof technique that can be used to prove theorems in various branches of mathematics. However, its specific application may vary depending on the context and concepts being studied.
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