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_09” 2011/5/22 23:21 page 25 #25
Sequences and Series of Functions 9-25
(a) {f
n
} is uniformly bounded on K.
(b) {f
n
} admits a subsequence that converges uniformly on K.
Proof The proof depends on the following steps:
Step 1: {f
n
} is uniformly bounded on K.
Step 2: There exists a countable dense subset E of K.
Step 3: {f
n
} admits a subsequence, say {f
n
k
= g
k
}, which converges
pointwise at every x E.
Step 4: {g
k
} converges uniformly on K.
Note that Step 1 and Step 4 are precisely the required conclusions.
We shall prove Step 1 independently and show that Step 2 and Step 3
together imply Step 4. Finally, we shall prove Step 2 and Step 3 to
complete the proof.
Proof of Step 1: Using equicontinuity of the family {f
n
}, given >0
we choose ...
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