Question: ............ (2) Let X be a set, and (fn) be a sequence of functions fn: X - R. Prove that if (fn) converges to some

............

(2) Let X be a set, and (fn) be a sequence of functions fn: X - R. Prove that if (fn) converges to some function f: X - R uniformly, then (fn) is uniformly Cauchy

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer

Step: 1 Unlock blur-text-image

Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock

Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Can you please answer 16,20,30,31,41 Thankyou. 18. Prove that the set of noneegative rational sambers is denumerable Subtraction on the set of nataral numbers is defined by sets is countable Prove...

Answer the questions. in the following attachments. Question 1. (10 marks) Consider the function z = f(x, y) = x/y. Showing all steps, use the Newton Quotient approach to find ?z/?y. Question 2. (5...

Let A CR, let (fn) be a sequence of functions from A to R, and let h R R be a continuous function. Show that, if (fn) con- verges pointwise to a function f, then the sequence (ho fn) converges...

Problem 2. Let X, Y, Z and suppose f: X Y and g : Y Z are functions. Consider the composition h = gof: X Z defined by h(r) = 9(f(x)). (a) Prove that if both f : X Y and g : Y Z are injective...

.Provide solutions to the following questions. Part B (10points, questions 11 to 20) Listed in the table below are National Accounts data for a small country: S (billions) Consumption 400 Imports 10...

Let fn () be a sequence of bounded real valued functions that converges uniformly to a function f(x) on S. Prove that if f(@) is bounded on S, then there exists some uniform bound M >0 such that Ifn...

Let E be a nonempty subset of R and f be a real-valued function defined on E. Suppose that fn is a sequence of bounded functions on E which converges to f uniformly on E. Prove that uniformly on E as...

5. Suppose all functions (fn) in a sequence of functions on A Q R are bounded. a) Show that, if (fn) converges to f uniformly on A, then f is bounded as well. b) Give an example Where (fn) converges...

Consider the measure space ( X , , ) , ( X ) . Suppose that f , f n , g , g n : X R , n N are measurable , i.e, a R : f 1 ( { a } ) ,where isa algebra . Prove that a) ( { x X f ( x ) > n } ) 0 , n ....

A company is considering an investment project with the following cash flows: Year 1: -$50,000 (initial investment) Year 2: $20,000 Year 3: $25,000 Year 4: $30,000 Calculate the payback period and...

On January 8, the end of the first weekly pay period of the year, Regis Companys payroll register showed that its employees earned $ 22,760 of office salaries and $ 65,840 of sales salaries....

You observe that the current spot rate for the Brazilian real ( BRL ) is 1 B R L = 0 . 1 6 8 8 USD. From the Brazilian perspective, this is a ( n ) quotation for the

Chinese Wall model: consider following COIs and CDs: Bank COI class: Chase Bank, US Bank, Citi Bank Gas Company COI class: Shell, BP, ARCO, Union76 Beverage class: Coca cola, Pepsi a. Assume we have...