Sequences of FunctionsDef. Let (fn) be a sequence of functio

Sequences of FunctionsDef. Let (fn) be a sequence of functions defined on a subset S of R. Then (fn) converges uniformly on S to a function f defined on S ifFor each e > 0 there exists a number N such that for all x in S, for all n > N , | fn (x) -f(x) | < e.#4.Let for x Î Let f (x) = 0.Complete the following discussion and proof that (fn) converges uniformly to f on .Discussion:Suppose e is any positive real number.We want to find N such that for all x Î and n > N, we have |fn(x) -f (x)| =< eNote that since x ³ we haveand £ ____.( ____are numbers in simplest form)______for all x Î(___ is an expression involving an appropriate constant and the variable n only, no x)So, we want______< e, which implies that n > _____.Proof:Let > 0. Choose N = _____. For all x Î , and n > N, we have______ <_______= e, as desired.(______ should be the expression involving e, before being simplified to get exactly e.)#5.Let for x Î R.Let f(x) = 3×2.Clearly, (fn) converges pointwise to f. But does it converge uniformly to f?Fill in the blanks to carefully show that (fn) does not converge uniformly to f onR.We must show: (the negation of the definition)For _____ (all/some) e > 0, for _____ (all/some) N, for_____ (all/some) x in Rand _____ (all/some) n> N ,| fn (x) – f(x) | __(<,>,£, ³)e.Let = 1. Given any N , let n be a positive integer greater than N, and setx = en.Then we have | fn (x) – f(x) | =______________________________________ (<,>,£, ³)1 = e.(NOTE: In the _____________________substitute forfn (x) and f(x) and simplify, applying x = en.)#6.Let for x Î [0, 1].#6(a) State f (x) = lim fn(x).#6 (b) Determine whether (fn) converges uniformly to f on[0, 1].Carefully justify your answer, either showing that the definition (shown on the previous page) holds, or proving that it cannot hold.#7.Letfor x Î [-0.8, 1].#7 (a) State a formula for f (x) = lim fn(x).(no explanation required)#7 (b)(fn) does not converge uniformly to f on [-0.8,1].How can you deduce this fact without working hard? Please explain. HINT: Apply an appropriate theorem and an observation about the function f(x) you found in part (a).Series of Functions (#8, 12 pts)#8. Determine whether or not the given series of functions converges uniformly on the indicated set. Justify your answers with work/explanations . (HINT: Consider the Weierstrass M-Test and find an appropriate sequence Mn )#8 (a)for x in R.#8(b)for x Î.#9.Show that it is possible to have a sequence of functions that are discontinuous at every point yet converge uniformly to a function that is continuous everywhere.That is, state an example of a sequenceof functions (fn)and a function f satisfying all of the following:Each fn is discontinuous at every real number.(fn)converges uniformly to f .f is continuous at every real number.(explanation not required)

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