# Sequences of Functions Def. Let (fn) be a sequence of funct

Sequences of Functions Def. 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 if For 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)| =< e Note 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)

### "Looking for a Similar Assignment? Order now and Get a Discount!

Our guarantees
1. High-Class Quality.
‘Will you write my paper for me that meets all requirements?’ This question is frequently asked by many students, and we always answer in the affirmative. Our main goal is to deliver a perfectly written paper the meets high writing standards. We don’t rest unless you are satisfied with our work. If you hire a paper writer online, we guarantee you that you get 100% original and plagiarism-free assignments of high quality.
2. Complete Anonymity.
We value your privacy and use modern encryption systems to protect you online. We don’t collect any personal or payment details and provide all our customers with 100% anonymity. ‘Can you write a paper for me so that I could stay anonymous?’ Of course, we can! We are here to help you, not to cause problems.
3. Fast Delivery.
We completely understand how strict deadlines maybe when it comes to writing your paper. Even if your paper is due tomorrow morning, you can always rely on us. Our writers meet all set deadlines unequivocally. This rule is ironclad! The offered range is wide and starts from 6 hours to 2 weeks. Which one to choose is totally up to you. For our part, we guarantee that our writers will deliver your order on time.