Uniform Convergence Homework: Is fn(x) Convergent on [a,b] & R?

[Kate2010]

Homework Statement

f_n(x) = 0 if x \leq n
= x-n if x\geq n

Is f_n(x) uniformly convergent on [a,b]?
Is it uniformly convergent on R?

Homework Equations

The Attempt at a Solution

I think lim_{n -> infinity} f_n(x)= 0
However I do not know what sup_{x in [a,b]} |f_n(x)| would be, I think it could be 0 but I don't know how to show this.

I'm unsure about over R as well?

 

[owlpride]

Have you looked at a couple of specific functions?

On an interval [a,b], the sequence f_n is eventually constant, isn't it?

 

[Kate2010]

I have maybe shown that it is uniformly convergent on the interval [a,b] but I still don't know about over the real numbers. My instinct is that it isn't but I'm not sure how to show this.

 

[owlpride]

It is not.

What is your definition of uniform convergence? Here's the one I learned: A sequence of functions f_n converges to f uniformly if for any eps>0, there exists an N such that |f_n(x) - f(x)|< eps for all n>N and x in the domain of f.

To show that a sequence does not converge uniformly (after showing it does converge pointwise), you could give yourself an arbitarily large n and show that |f_n(x) - f(x)| > 1 for some x depending on n.

 

[Kate2010]

I've been using:

f_n converges to f uniformly if there exists an N, for all n>N, m_n := sup|f_n(x) - f(x)| exists and tends to 0 as n tends to infinity.

I thought that on the interval [a,b] f_n(x) and f(x) would equal 0 as n tends to infinity?

 

[owlpride]

I thought that on the interval [a,b] f_n(x) and f(x) would equal 0 as n tends to infinity?

Correct. Explicitly, if n > b, then f_n(x) = 0 for all x in [a,b].

f_n converges to f uniformly if there exists an N, for all n>N, m_n := sup|f_n(x) - f(x)| exists and tends to 0 as n tends to infinity.

Let's start with "sup|f_n(x) - f(x)| exists". You can show that f_n -> 0 pointwise, correct? Pick your favorite number n. Does sup_x |f_n(x) - 0| exist?

 

[Kate2010]

So, say n=10, then sup | f_10(x) - 0 | = sup | x - 10 | = b-10 ?

 

[rsa58]

the issue here is the set over which we are looking at the function. at some point n=N will be bigger than b so f will be zero for all x in this interval for all values of n>=N. hence uniform convergence. for the second one have you heard of the cauchy criterion for uniform convergence? look it up. notice that when you subtract different functions from the sequence they leave an integer remainder. this does not get arbitrarily small...

 

[owlpride]

So, say n=10, then sup | f_10(x) - 0 | = sup | x - 10 | = b-10 ?

Yes, but only if your function is restricted to an interval [a,b] with b > 10. What happens when you let your function vary over the entire real line? Is there an upper bound on the difference |f_10(x)-0| = x-10 if you let x get arbitrarily large?

 

[Kate2010]

Thanks, I think I understand now :)

 

[rsa58]

also the sequence x-n is simply a shifting down the y-axis of the function y=x. also it intersects the x-axis at x=n. this sequences is essentially going move out at a constant velocity in the x direction. these lines continue out to infinity without getting arbitrarily close to one another nor close to any concievable function.