Q5 Real analysis by Walton rudinIt's solution already sent by u but not cleared .plzz send handwritten in short mathematical form not more there.... hence lies in B. If f is a complex continuous function on K, f = u +iv,then u e B, v e B, hence fe B. This completes the proof.EXERCISES1. Prove that every uniformly convergent sequence of bounded functions is uni-formly bounded.2. If (f} and {gn} converge uniformly on a set E, prove that {fa+ g} convergesuniformly on E. If, in addition, {fa} and {g„} are sequences of bounded functions,prove that {fugn} converges uniformly on E.3. Construct sequences {fa}, {g,} which converge uniformly on some set E, but suchthat {fng) does not converge uniformly on E (of course, {fa9a} must converge onE).4. Considerf(x) =For what values of x does the series converge absolutely? On what intervals doesit converge uniformly? On what intervals does it fail to converge uniformly? Is fcontinuous wherever the series converges? Is f bounded?166 PRINCIPLES OF MATHEMATICAL ANALYSIS5. Let10f.(x)={ sin?Show that {f} converges to a continuous function, but not uniformly. Use theseries E f, to show that absolute convergence, even for all x, does not imply uni-form convergence.6. Prove that the seriesconverges uniformly in every bounded interval, but does not converge absolutelyfor any value of x.7. For n= 1, 2, 3, ..., x real, put1+ nx²Show that {f.) converges uniformly to a function f, and that the equationf'(x)= lim f:(x)is correct if x * 0, but false if x = 0.8. If(x<0),(x>0),I(x) =if (x,) is a sequence of distinct points of (a, b), and if E c. converges, prove thatthe seriesf(x) =- x,)(aSxsb)converges uniformly, and that f is continuous for every x*x.9. Let {f.) be a seguence of continuous functions which converges uniformly to a

For a sequence of functions, there are two different types of convergence. These are point wise…

166 PRINCIPLES OF MATHEMATICAL ANALYSIS 5. Let (). n+ S(x)={sin? Show that {f) converges to a continuous function, but not uniformly. Use the series E f, to show that absolute convergence, even for all x, does not imply uni- form convergence.

166 PRINCIPLES OF MATHEMATICAL ANALYSIS 5. Let (). n+ S(x)={sin? Show that {f) converges to a continuous function, but not uniformly. Use the series E f, to show that absolute convergence, even for all x, does not imply uni- form convergence.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.1: Infinite Sequences And Summation Notation

Problem 74E

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Question

Q5 Real analysis by Walton rudin It's solution already sent by u but not cleared .plzz send handwritten in short mathematical form not more there....

hence lies in B. If f is a complex continuous function on K, f = u +iv,
then u e B, v e B, hence fe B. This completes the proof.
EXERCISES
1. Prove that every uniformly convergent sequence of bounded functions is uni-
formly bounded.
2. If (f} and {gn} converge uniformly on a set E, prove that {fa+ g} converges
uniformly on E. If, in addition, {fa} and {g„} are sequences of bounded functions,
prove that {fugn} converges uniformly on E.
3. Construct sequences {fa}, {g,} which converge uniformly on some set E, but such
that {fng) does not converge uniformly on E (of course, {fa9a} must converge on
E).
4. Consider
f(x) =
For what values of x does the series converge absolutely? On what intervals does
it converge uniformly? On what intervals does it fail to converge uniformly? Is f
continuous wherever the series converges? Is f bounded?
166 PRINCIPLES OF MATHEMATICAL ANALYSIS
5. Let
10
f.(x)={ sin?
Show that {f} converges to a continuous function, but not uniformly. Use the
series E f, to show that absolute convergence, even for all x, does not imply uni-
form convergence.
6. Prove that the series
converges uniformly in every bounded interval, but does not converge absolutely
for any value of x.
7. For n= 1, 2, 3, ..., x real, put
1+ nx²
Show that {f.) converges uniformly to a function f, and that the equation
f'(x)= lim f:(x)
is correct if x * 0, but false if x = 0.
8. If
(x<0),
(x>0),
I(x) =
if (x,) is a sequence of distinct points of (a, b), and if E c. converges, prove that
the series
f(x) =
- x,)
(aSxsb)
converges uniformly, and that f is continuous for every x*x.
9. Let {f.) be a seguence of continuous functions which converges uniformly to a

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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