Real Analysis II is a continuation of Real Analysis I. In this document you will find the following :
- Countable and Uncountable Sets
- Topology of Real Line
- Uniform Continuity
- Riemann Integration
- Uniform Convergence
When we count the elements in a set, we say “one, two, three, . . . ,” stopping when we
have exhausted the set. From a mathematical perspective, what we are doing is defining
a bijective mapping between the set and a portion of the set of natural numbers. If the
set is such that the counting does not terminate, such as the set of natural numbers
itself, then we describe the set as being infinite.
The notions of “finite” and “infinite” are extremely primitive, and it is very likely that
the reader has never examined these notions very carefully. In this section we will define
these terms precisely and establish a few basic results and state some other important
results that seem obvious but whose proofs are a bit tricky.
Definition 1.1.1. (a) The empty set ∅ is said to have 0 elements.
(b) If n ∈ N, a set S is said to have n elements if there exists a bijection from the set
Nn := {1, 2, . . . , n} onto S.
(c) A set S is said to be finite if it is either empty or it has n elements for some
n ∈ N.
(d) A set S is said to be infinite if it is not finite.
1
, 1.2 Countable Sets
We now introduce an important type of infinite set.
Definition 1.2.1. (a) A set S is said to be denumerable (or countably infinite) if
there exists a bijection of N onto S.
(b) A set S is said to be countable if it is either finite or denumerable.
(c) A set S is said to be uncountable if it is not countable.
From the properties of bijections, it is clear that S is denumerable if and only if there
exists a bijection of S onto N. Also a set S1 is denumerable if and only if there exists
a bijection from S1 onto a set S2 that is denumerable. Further, a set T1 is countable if
and only if there exists a bijection from T1 onto a set T2 that is countable. Finally, an
infinite countable set is denumerable.
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through EFT, credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying this summary from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller keanuperumal. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy this summary for R79,00. You're not tied to anything after your purchase.
