Q 8 solve real analysis by Walton rudin 166 PRINCIPLES OF MATHEMATICAL ANALYSIS5. Letf.(x)={ sin+1(-)Show that {f} converges to a continuous function, but not uniformly. Use theseries E fa 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, putf.(x) =niShow that {f} converges uniformly to a function f, and that the equationf'(x) = limf(x)is correct if x #0, but false if x=0.8. If1(х) —(xS0),(x 0),if (x.) is a sequence of distinct points of (a, b), and if E|ca converges, prove thatthe seriesS(x) = c. I(x – x)(asxsb)converges uniformly, and that f is continuous for every x + x,.9. Let {fa) be a sequence of continuous functions which converges uniformly to afunction f on a set E. Prove thatlim f.(x,) = f(x)for every sequence of points x, E E such that xn +x, and x e E. Is the converse ofthis true?SEQUENCES AND SERIES OF FUNCTIONS 16710. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4,for the definition), consider the function(пх)f(x) =(x real).Find all discontinuities of f, and show that they form a countable dense set.Show that fis nevertheless Riemann-integrable on every bounded interval.11. Suppose (fa), (e) are defined on E, and(a) Ef, has uniformly bounded partial sums;(b) g. -0 uniformly on E;(c) g,(x) 292(x)2gs(x) 2 for every x e E.Prove that E fng, converges uniformly on E. Hint: Compare with Theorem3.42.11 Sunnone a ond f (.Lare defined on (0 m) are Riemann-integrable on

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Q 8 solve real analysis by Walton rudin 166 PRINCIPLES OF MATHEMATICAL ANALYSIS5. Letf.(x)={ sin+1(-)Show that {f} converges to a continuous function, but not uniformly. Use theseries E fa 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, putf.(x) =niShow that {f} converges uniformly to a function f, and that the equationf'(x) = limf(x)is correct if x #0, but false if x=0.8. If1(х) —(xS0),(x >0),if (x.) is a sequence of distinct points of (a, b), and if E|ca converges, prove thatthe seriesS(x) = c. I(x – x)(asxsb)converges uniformly, and that f is continuous for every x + x,.9. Let {fa) be a sequence of continuous functions which converges uniformly to afunction f on a set E. Prove thatlim f.(x,) = f(x)for every sequence of points x, E E such that xn +x, and x e E. Is the converse ofthis true?SEQUENCES AND SERIES OF FUNCTIONS 16710. Letting (x) denote the fractional part of the real number x (see Exercise 16, Chap. 4,for the definition), consider the function(пх)f(x) =(x real).Find all discontinuities of f, and show that they form a countable dense set.Show that fis nevertheless Riemann-integrable on every bounded interval.11. Suppose (fa), (e) are defined on E, and(a) Ef, has uniformly bounded partial sums;(b) g. ->0 uniformly on E;(c) g,(x) 292(x)2gs(x) 2 for every x e E.Prove that E fng, converges uniformly on E. Hint: Compare with Theorem3.42.11 Sunnone a ond f (.Lare defined on (0 m) are Riemann-integrable on

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8. If (x) = 0 (xS0), (x>0), if (x.) is a sequence of distinct points of (a, b), and if Elc.| converges, prove that the series S«) = Ža. (x – x) (asxsb) converges uniformly, and that fis continuous for every x + x..