Given the reals a<b find a, t such that :

 

For all r>0 there exist an , x such that : 0<|x-t|<r and a<x<b

x=a + epsilon

t=a

 

epsilon is very small

x=a + epsilon

t=a

 

epsilon is very small

If you put t=a and x=a+ε ,then the inequality becomes: |a+ε-a|<ε => ε<ε, a contradiction

 

By the way in the OP it is r and not ε

If you put t=a and x=a+ε ,then the inequality becomes: |a+ε-a|<ε => ε<ε, a contradiction

 

By the way in the OP it is r and not ε

"By the way in the OP it is r and not ε" - exactly and for the inquality to work epsilon is smaller than r (ie now matter hw small r is there is a smaller real that we designate epsilon)

 

the inequality becomes

0<|a+ε-a|<r =>

0<ε<r

"By the way in the OP it is r and not ε" - exactly and for the inquality to work epsilon is smaller than r (ie now matter hw small r is there is a smaller real that we designate epsilon)

 

the inequality becomes

0<|a+ε-a|<r =>

0<ε<r

Suppose r=3b-a

And since you clame the epsilon that you introduce in the problem to be ε<r then we can say that ε can be equal to 2b-a<3b-a=r

Hence if we put

x=a+ε=a+(2b-a)=2b

Does this x satisfy also the relation a<x<b ?

Furthermore when we introduce a new variable into a problem we have to quantify that variable to make our formula a well formed formula. And since you introduce ε how do you quantify it?

Suppose r=3b-a

And since you clame the epsilon that you introduce in the problem to be ε<r then we can say that ε can be equal to 2b-a<3b-a=r

Hence if we put

x=a+ε=a+(2b-a)=2b

Does this x satisfy also the relation a<x<b ?

Furthermore when we introduce a new variable into a problem we have to quantify that variable to make our formula a well formed formula. And since you introduce ε how do you quantify it?

 

No - if you checked my first response I said that epsilon was very small. Sorry I had thought you would know this terminology; epsilon is used to denote a very small addition of arbitrary smallness.

 

What I am saying in my posts is that x is the tiniest bit bigger than a and that a is equal to t

 

 

http://en.wikipedia.org/wiki/Epsilon

 

In mathematics (particularly calculus), an arbitrarily small positive quantity is commonly denoted ε; see (ε, δ)-definition of limit.

• In reference to this, the late mathematician Paul Erdős also used the term "epsilons" to refer to children (Hoffman 1998, p. 4).

 

No - if you checked my first response I said that epsilon was very small. Sorry I had thought you would know this terminology; epsilon is used to denote a very small addition of arbitrary smallness.

 

What I am saying in my posts is that x is the tiniest bit bigger than a and that a is equal to t

 

 

http://en.wikipedia.org/wiki/Epsil

In the whole of mathematics can you show me a single definition defining the exprssion " small epsilon"?

In the definition of limits we say for all epsilon biger tha 0 ,there exists a delta biger than 0 such that: e.t.c e.t.c

99% of the problems in analysis stert with the expession . Let ε>0, or given ε>0

Yes you can use small or big epsilon,or delta .

But if you do not clearly show their relatin to other variables of the problem or between them it is catastrophic.

In the limits for example ,you have to clearly show the relatio9n between epsilon and delta, whether the epsilon and deltas are small or big,or whatever

In your solution what will happen if for your " small epsion" we have a small : b-a??

Besides when you say : " for a small epsilon ",one may say ,ok But how small .

Mathematics is relations between variables and certainly expressions or words of : small,big ,tiny e.t.c do not show relations

Edited April 26, 201510 yr by triclino

But it's clear what the relationship must be. We must have [math]\varepsilon < \min(r, b - a)[/math]. If you want an equality, use the fact that [math]\min(x, y) = \frac{x + y - |x - y|}{2}[/math].

 

In the whole of mathematics can you show me a single definition defining the exprssion " small epsilon"?

In the definition of limits we say for all epsilon biger tha 0 ,there exists a delta biger than 0 such that: e.t.c e.t.c

99% of the problems in analysis stert with the expession . Let ε>0, or given ε>0

Yes you can use small or big epsilon,or delta .

But if you do not clearly show their relatin to other variables of the problem or between them it is catastrophic.

In the limits for example ,you have to clearly show the relatio9n between epsilon and delta, whether the epsilon and deltas are small or big,or whatever

In your solution what will happen if for your " small epsion" we have a small : b-a??

Besides when you say : " for a small epsilon ",one may say ,ok But how small .

Mathematics is relations between variables and certainly expressions or words of : small,big ,tiny e.t.c do not show relations

 

 

http://mathworld.wolfram.com/Epsilon.html

 

It is a little disingenuous to put "small epsilon" in quotation marks as if it is a phrase I have used - the only person to have used the phrase small epsilon is you.

 

"99% of the problems..." really?

 

"In your solution what will happen if for your " small epsion" we have a small : b-a??" It is arbitrarily small - ie it is smaller than... John even quantified the maximum above.