Positive derivative implies growing function using Bolzano-Weierstrass

In summary: Then, we can use the BWT to show that there exists a convergent subsequence ##x_{n_k}## that converges to a point ##c##. Since the function is continuous, we know that ##f(x_{n_k})\to f(c)## as ##n_k\to\infty##. But since ##f(x)<f(y)## for all ##x,y\in[a,b]## with ##x<y##, it must be the case that ##f(c)\leq f(x_{n_k})<f(y_{n_k})\leq f(y)##. This contradicts our assumption that
  • #1
schniefen
178
4
Homework Statement
Using the Bolzano-Weierstrass theorem (for every bounded sequence, there exists a convergent subsequence), prove that a positive derivative implies a growing function.
Relevant Equations
The Bolzano-Weierstrass theorem: Every bounded sequence ##c_n## has a convergent subsequence ##c_{n_k}##.
I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement:
$$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the interval. I've been instructed to proceed with a proof by contradiction, yet am slightly confused over how one would start such a proof. I would assume ##f'(x)>0## over a given interval but then also assume the logical negotiation of what that implies, correct? In other words, that the function is then decreasing over the interval and show that this leads to a contradiction. I can also somewhat see how one would go about proving it... ##\exists \ x_3 \in[x_1,x_2]## ...but don't see how the Bolzano Weierstrass theorem comes into play.
 
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  • #2
I don't see a quick way to prove the claim using Bolzanno Weierstrass theorem. Are you familiar with the mean value theorem? It provides a proof that takes 2 lines...
 
  • #3
schniefen said:
Problem Statement
Using the Bolzano-Weierstrass theorem (for every bounded sequence, there exists a convergent subsequence), prove that a positive derivative implies a growing function.

How do you define a "growing function"?
 
  • #4
PeroK said:
How do you define a "growing function"?

It's in the quantor statement of the theorem.
 
  • #5
Math_QED said:
I don't see a quick way to prove the claim using Bolzanno Weierstrass theorem. Are you familiar with the mean value theorem? It provides a proof that takes 2 lines...

Yes, I am familiar with the mean value theorem and the proof using it. This time I've been asked to use the Bolzano Weierstrass theorem instead of the mean value theorem.
 
  • #6
Math_QED said:
It's in the quantor statement of the theorem.

I'll leave it to you then, as I have no idea what that means.
 
  • #7
PeroK said:
How do you define a "growing function"?
This is how I would define a growing function over a given interval ##[a,b]##: ## \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2##
 
  • #8
schniefen said:
This is how I would define a growing function over a given interval ##[a,b]##: ## \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2##

Okay, that's strictly monotonic increasing in my terminology!
 
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  • #9
PeroK said:
Okay, that's strictly monotonic increasing in my terminology!

In mine as well, but there does not seem to be consensus sadly. I use the term "non-decreasing" for what others call "increasing", lately.
 
  • #10
schniefen said:
Problem Statement
Using the Bolzano-Weierstrass theorem (for every bounded sequence, there exists a convergent subsequence), prove that a positive derivative implies a growing function.
Relevant Equations
The Bolzano-Weierstrass theorem: Every bounded sequence ##c_n## has a convergent subsequence ##c_{n_k}##.

I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement:
$$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the interval. I've been instructed to proceed with a proof by contradiction, yet am slightly confused over how one would start such a proof. I would assume ##f'(x)>0## over a given interval but then also assume the logical negotiation of what that implies, correct? In other words, that the function is then decreasing over the interval and show that this leads to a contradiction. I can also somewhat see how one would go about proving it... ##\exists \ x_3 \in[x_1,x_2]## ...but don't see how the Bolzano Weierstrass theorem comes into play.

You need to generate a sequence in some way. Then, use the BWT to show that the sequence converges to a point. Then, conclude something about that point.

You could start with your positive derivative on ##[a, b]##. Then, assume there exist ##x < y \in [a, b]## with ##f(x) > f(y)##.

Now, what next?
 
  • #11
schniefen said:
Yes, I am familiar with the mean value theorem and the proof using it. This time I've been asked to use the Bolzano Weierstrass theorem instead of the mean value theorem.
The mean value theorem is a corollary from Rolle. Thus the question reduces to: how to prove Rolle with Bolzano-Weiertsraß? We already have a bounded interval, so all what is left to do is, to find a sequence of the form ##\frac{f(a)-f(b)}{a-b}## converging to zero.
 
  • #12
PeroK said:
You need to generate a sequence in some way. Then, use the BWT to show that the sequence converges to a point. Then, conclude something about that point.

You could start with your positive derivative on ##[a, b]##. Then, assume there exist ##x_1 < y_1 \in [a, b]## with ##f(x_1) > f(y_1)##.

Now, what next?

... what about looking at the point ##\frac{x_1+y_1}{2}##?
 
  • #13
PeroK said:
You need to generate a sequence in some way. Then, use the BWT to show that the sequence converges to a point. Then, conclude something about that point.

You could start with your positive derivative on ##[a, b]##. Then, assume there exist ##x < y \in [a, b]## with ##f(x) > f(y)##.

Now, what next?
This feels like the right outline of the proof, but I can't see what we're looking for in the proof. What are we looking for in the point that a possible subsequence would converge to?
 
  • #14
schniefen said:
This feels like the right outline of the proof, but I can't see what we're looking for in the proof. What are we looking for in the point that a possible subsequence would converge to?

I can see how to get a proof but it doesn't actually need the BWT per se. And I'm not sure how enlightening it would be.

Maybe there's a slick way using the BTW, but I don't see it.
 
  • #15
schniefen said:
I'm stuck on a proof involving the Bolzano-Weierstrass theorem. Consider the following statement:
$$f'(x)>0 \ \text{on} \ [a,b] \implies \forall x_1,x_2\in[a,b], \ f(x_1)<f(x_2) \ \text{for} \ x_1<x_2 $$ i.e. a positive derivative over an interval implies that the function is growing over the interval. I've been instructed to proceed with a proof by contradiction, yet am slightly confused over how one would start such a proof.
Since people are stuck, I'll offer an approach that precedes Rolle's.

However, it's still a little abusive on my part, because what I'm doing here is only closely related to what OP asks for-- in essence the below is a compactness argument. (i.e. in this case it's really the extreme value theorem which is frequently proven in short order with Bolzano Weierstrass.)
- - - - -
It's enough to consider two cases: either

##\text{(i)} f(a) \lt f(x)## for ##x \in (a,b]## i.e. the left boundary is mapped to the the strict minimum in this closed set
##\text{(ii)}## there is (at least one) ##x^* \in(a,b]## where ## f(x^*) \leq f(x)## for all ##x \in [a,b]##.

in the case of ##\text{(i)}## everything works great, so you can skip to the end.
But ##\text{(ii)}## isn't true so proceed by contradiction: Consider the sequence of left difference quotients at ##f(x^*)## we know they converge to ##f'(x^*) = c\gt 0##, select ##\epsilon = \frac{c}{2}## and we find a contradiction if ##f(x^*)## is a minimum.

the fact the ##f## is differentiable implies continuity, which combined with compactness guarantees a minimum value of ##f## in this ##[a,b]##, thus we must have ##\text{(i)}##.

with this result in hand, consider re-running the argument on a particular subset, i.e. ##[x_1, x_2]##
 

Related to Positive derivative implies growing function using Bolzano-Weierstrass

1. What is the Bolzano-Weierstrass theorem?

The Bolzano-Weierstrass theorem is a fundamental theorem in calculus that states that any bounded sequence of real numbers has a convergent subsequence. In other words, if a sequence is contained within a finite interval, there will always be a subsequence that converges to a limit within that interval.

2. How is the Bolzano-Weierstrass theorem related to positive derivatives and growing functions?

The Bolzano-Weierstrass theorem is often used to prove that a function is increasing or decreasing over a certain interval. If a function has a positive derivative at every point in that interval, it means that the function is increasing and the Bolzano-Weierstrass theorem guarantees the existence of a convergent subsequence within that interval.

3. Can you give an example of how the Bolzano-Weierstrass theorem can be used to prove a function is increasing?

Yes, for example, let's say we have a function f(x) = x^2 on the interval [0,1]. We can show that the derivative of this function, f'(x) = 2x, is always positive on this interval. By the Bolzano-Weierstrass theorem, we know that there exists a convergent subsequence within this interval, which means that the function is increasing on this interval.

4. Is the Bolzano-Weierstrass theorem only applicable to real-valued functions?

No, the Bolzano-Weierstrass theorem can also be applied to complex-valued functions. In this case, the sequence must be contained within a bounded subset of the complex plane, and the convergent subsequence will also converge to a complex number within that subset.

5. Are there any other applications of the Bolzano-Weierstrass theorem?

Yes, the Bolzano-Weierstrass theorem has many applications in mathematics, including in real analysis, topology, and functional analysis. It is also used in various fields of science, such as physics and economics, to prove the existence of solutions to certain problems.

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