Skip to Main Content
Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_04” 2011/5/22 23:17 page 16 #16
4-16 Real Analysis
points of G, then I
x
and I
y
are either disjoint or identical. Indeed, if they
have a common point z, then I
x
= I
z
and I
y
= I
z
and hence I
x
= I
y
.
Consider the class I of all distinct sets of the form I
x
for points x in G
(note that I may be a finite of infinite collection). This is a disjoint class
of open intervals and G is obviously its union. It remains to prove that
I is atmost countable. Let G
r
be the set of rational points in G. G
r
is
clearly non-empty. We define a mapping f of G
r
onto I as follows: for
each s in G
r
, let f (s) be that unique interval in I , which contains r. G
r
is countable and hence I is atmost countable (Note that f is onto and
from Theorem 2.5.4(ii) we can deduce ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Make: Calculus

Make: Calculus

Joan Horvath, Rich Cameron
Complex Analysis

Complex Analysis

ITL Education

Publisher Resources

ISBN: 9781299447561Publisher Website