# How to prove a set is unbounded

by happybear
Tags: prove, unbounded
 P: 19 1. The problem statement, all variables and given/known data Suppose I have a set E that contains all continuous function f from [a,b] --> R, I think this is unbounded, but can I prove it? 2. Relevant equations d(f1,f2)= sup{f1(x)-f2(x)} 3. The attempt at a solution I want to show |d(f1,f2)|>M for some M, but I don't know if this is the right direction and how to do it
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 A set is "unbounded" if, for any positive number A, there exist point p and q in the set such that d(p,q)> A. That, of course, depends on how you measure "distance"- d(p,q). What is your definition of d(f, g) for f and g in the set of continuous functions from [a, b] to R? (What ever your d(f,g) is, consider the functions fn(x)= n. for n= 1, 2, 3, ...)
 P: 19 But there are no definition of f, that is the point. F will be any continuous function from [a,b] to R. Can I prove this set is infinite?
 P: 238 How to prove a set is unbounded If you think that it is false, you can provide a counterexample. So, if $$f : [a, b] \to R$$ and let A = [a, b] is bounded, is it necessarily true that f(A) is bounded?
Math
Emeritus
Thanks
PF Gold
P: 39,682
 Quote by happybear But there are no definition of f, that is the point. F will be any continuous function from [a,b] to R. Can I prove this set is infinite?
Do you mean d, the distance function? There has to be a definition of distance in order to talk about "bounded" or "unbounded". And you don't want to prove the set is infinite (that's trivial), you want to prove it is unbounded, a completely different thing.

Is it possible that you are talking about the set of continuous functions from [a,b] to R? For that, the distance function is most commonly $d(f, g)= \max_{a\le x\le b} |f(x)- g(x)|$.
Math
Emeritus
Thanks
PF Gold
P: 39,682
 Quote by konthelion If you think that it is false, you can provide a counterexample. So, if $$f : [a, b] \to R$$ and let A = [a, b] is bounded, is it necessarily true that f(A) is bounded?
The problem was not to prove that functions in the set are bounded but that the set itself is.
P: 19
 Quote by HallsofIvy Is it possible that you are talking about the set of continuous functions from [a,b] to R? For that, the distance function is most commonly $d(f, g)= \max_{a\le x\le b} |f(x)- g(x)|$.

Yes. That is what I meant. And what I want to show is that there are infinitely many functions of f. That is how I interpret the question, since they ask if the set is unbounded. Or do I misunderstand it

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