An existence proof shows that an object exists. In some cases, this means displaying the object, or giving a method for finding it.
Example. Show that there is a real number x such that but .
There are many possibilities; for example,
In some cases, you can know that an object exists without having any way of finding it (or finding it exactly). By analogy:
You've seen results of this kind in calculus. One such result is the Intermediate Value Theorem:
Example. Show that there is a real number x such that .
The assertion means that the graphs of and intersect:
It looks like they do. Note, however, that a picture is not a proof.
Let . Then
Since is positive and is negative, and since f is continuous for all x, the Intermediate Value Theorem implies that there is an x between 0 and for which . Then , so .
Notice that the Intermediate Value Theorem doesn't tell you what x is, or how to find it. (It's approximately 0.73909.)
To say that there is an x satisfying a certain property does not mean that there is only one x satisfying the property. If that is what is meant, it has to be stated explicitly.
Example. The Mean Value Theorem says that if f is function which is continuous on the closed interval and differentiable on the open interval , then
for some number c such that .
The result says that there is a number c. This doesn't mean that you might not have several c's that work.
For example, suppose and the interval is . Then
Now , so setting , I find that . Both of these values satisfy the conclusion of the Mean Value Theorem.
The definition of the limit is an example of an existence assertion.
Let f be a function from the real numbers to the real numbers, and let c be a real number. The statement
means:
\item{} For every , there is a , such that if , then .
Think of as a thermostat, as the actual temperature in a room, and L as the ideal temperature. Someone challenges you to make the actual temperature fall within a certain tolerance of the ideal temperature L. You must do that by setting your -thermostat appropriately (so that x is sufficiently close to c).
Moreover, note that it says "for every ". It's isn't enough for you to say what you'd do if you were challenged with or . You must prove that you can meet the challenge no matter what you're challenged with.
Finally, note the stipulation " ". This implies that , since gives . Thus, the conclusion " " must hold only for x's close to c, but not necessarily for . (It may hold for , but it doesn't have to.)
What does this mean? It's a precise way of saying that the value of the limit of as x approaches c does not depend on what does at --- over even whether is defined.
For example, consider the functions whose graphs are shown below.
In both cases,
In the first case, : The value of the function at is different from the value of the limit.
In the second case, is undefined.
The fact that means that f is not continuous at .
Example. Use the definition of the limit to prove that
In this case, , , and . So here is what I need to prove.
Suppose . I must find a such that if , then .
Note that at this point is fixed --- given --- but all you can assume is that it's some positive number. Since it is given, however, I can use it in finding an appropriate .
I'll show how to find by working backwards; then I'll write the proof "forwards", the way you should write it.
I want
It looks like I should set .
All of this has been on "scratch paper"; now here's the real proof.
Suppose . Let . If , then
Thus, if and , then . This proves that .
Example. Let
Use the definition of the limit to prove that
Let . I must find such that if , then .
Here's my scratch work. First, for ,
It looks like I should take .
For ,
It looks like I should take .
In order to ensure that both the and requirements are satisfied, I'll take to be the smaller of the two: .
Now here's the proof written out correctly.
Suppose . Let , and assume that .
If , then
Now consider the case . Since , and since , I have . Therefore,
(The case is ruled out because .)
Thus, taking guarantees that if , then . This proves that .
Example. Use the definition of the limit to prove that
Let . I want to find such that if , then .
I start out as usual with my scratch work:
Now I have a problem. I can use to control , but what do I do about ?
The idea is this: Since I have complete control over , I can assume . When I finally set , I can make it smaller if necessary to ensure that this condition is met.
Now if , then , so , and . In particular, the biggest could be is 5. So now
This inequality suggests that I set --- but then I remember that I needed to assume . I can meet both of these conditions by setting to the smaller of 1 and : that is, .
That was scratchwork; now here's the real proof.
Let . Set . Suppose .
Since , I have , so , or . Therefore, .
Now , so .
Now multiply the inequalities and :
Thus, if and , then . This proves that .
Example. Prove that .
Let . I must find such that if , then .
I'll start with some scratchwork.
I can use to control directly. I need to control the size of . It's important to think of this as , not as and !
Assume . Then , so .
For , , so .
For , , so , and .
Since all the number involved are positive, I can multiply the inequalities to obtain
Thus, I'll get if I have , or . Here's the proof.
Let . Set . Suppose .
Since , , and .
First, , so .
Next, , , so , and .
Hence,
In addition, . Therefore,
This proves that .
Copyright 2009 by Bruce Ikenaga