2. Suppose that fn converges uniformly to f on the interval I and that each fn is bounded on I, that is Mn = sup{|fn(x)| : x € I} < +∞ for each n € N. (i) Show that f is bounded on I. (ii) Show that there exists a constant M> 0 such that Mn ≤ M for all n € N.
2. Suppose that fn converges uniformly to f on the interval I and that each fn is bounded on I, that is Mn = sup{|fn(x)| : x € I} < +∞ for each n € N. (i) Show that f is bounded on I. (ii) Show that there exists a constant M> 0 such that Mn ≤ M for all n € N.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.5: The Kernel And Range Of A Linear Transformation
Problem 30EQ
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Question
![2. Suppose that fn converges uniformly to f on the interval I and that each fn is
bounded on I, that is
Mn = sup{|fn(x)| : x € I} < +∞
for each n € N.
(i) Show that f is bounded on I.
(ii) Show that there exists a constant M> 0 such that Mn M for all n € N.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F833bf7b1-3e6b-4749-8e88-54090320a3f5%2Fe0b0ea1f-f826-4808-86a9-64d67b2f494f%2Fqybyz9_processed.png&w=3840&q=75)
Transcribed Image Text:2. Suppose that fn converges uniformly to f on the interval I and that each fn is
bounded on I, that is
Mn = sup{|fn(x)| : x € I} < +∞
for each n € N.
(i) Show that f is bounded on I.
(ii) Show that there exists a constant M> 0 such that Mn M for all n € N.
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