In a course on introductory real analysis, Riemann integral is commonly defined through upper sums and lower sums, by which it is also known as the Darboux integral. Although the two definitions are equivalent, the concept of supremum and infimum are essential to define upper sums and lower sums, which explains why Darboux integral is commonly omitted from a Calculus course. Your task is to discuss about the concept of supremum and infimum for real numbers. Here are some hints on what your presentation can cover. 1. Give the definition of supremum and infimum for a nonempty subset of R. 2. Provide some examples to illustrate the concepts of supremum and infimum. 3. Discuss about the Completeness Axiom.
In a course on introductory
two definitions are equivalent, the concept of supremum and infimum are essential to define
upper sums and lower sums, which explains why Darboux integral is commonly omitted from
a Calculus course. Your task is to discuss about the concept of supremum and infimum for real
numbers. Here are some hints on what your presentation can cover.
1. Give the definition of supremum and infimum for a nonempty subset of R.
2. Provide some examples to illustrate the concepts of supremum and infimum.
3. Discuss about the Completeness Axiom.
4. Find some standard exercise pertaining to supremum and infimum and discuss its solution.
5. State and prove the Archimedean Property using the Completeness Axiom.
Step by step
Solved in 3 steps