Math Help - Proving espilon > x in S

1. Proving espilon > x in S

Hello,
Would someone please give me some clue as how I start to solve this problem: Let S = (0,1), Show that for each (epsilon) e > 0 there exist an x in S such that x < e (epsilon)
Thanks

2. Re: Proving espilon > x in S

Originally Posted by Mathlv
Would someone please give me some clue as how I start to solve this problem: Let S = (0,1), Show that for each (epsilon) e > 0 there exist an x in S such that x < e (epsilon)
If $\epsilon>0$ then $\frac{1}{\epsilon}>0$. Because the positive integers are not bounded above, $\exists N\in\mathbb{Z}^+\left[N>\frac{1}{\epsilon}\right].$

Can you finish?

3. Re: Proving espilon > x in S

I think it is sufficient to note that, for every $\epsilon> 0$, $0< \dfrac{\epsilon}{2}< \epsilon$.

(If $\dfrac{\epsilon}{2}> 1$ take x= 1/2.)

4. Re: Proving espilon > x in S

Plato,
Thank you for helping me. But I think I will try to continue from the clue of HallsofIvy below.

5. Re: Proving espilon > x in S

HallsofIvy,
Thank you for the help,
Here is what I have as a solution for the problem using your helping clue.

The Problem: Let S = (0,1)

Prove for each $\varepsilon$ > 0 there exist an x $\in$ S such that x < $\varepsilon$.

Solution:

For every $\varepsilon$ > 0, 0 < $\frac{\varepsilon }{2}$ < $\varepsilon$

Consider $\frac{\varepsilon }{2}$ and x = $\frac{1}{2}$
then

x < $\varepsilon$

$\Leftrightarrow$ $\frac{1}{2}$ < $\frac{\varepsilon }{2}$

$\Leftrightarrow$ 1 < $\varepsilon$

Thus

$\varepsilon$ > 0 and $\varepsilon$ = 1 > x = $\frac{1}{2}$

Therefore x < $\varepsilon$

Q.E.D

6. Re: Proving espilon > x in S

Originally Posted by Mathlv
HallsofIvy,
Thank you for the help,
Here is what I have as a solution for the problem using your helping clue.

The Problem: Let S = (0,1)

Prove for each $\varepsilon$ > 0 there exist an x $\in$ S such that x < $\varepsilon$.

Solution:

For every $\varepsilon$ > 0, 0 < $\frac{\varepsilon }{2}$ < $\varepsilon$

Consider $\frac{\varepsilon }{2}$ and x = $\frac{1}{2}$
then

x < $\varepsilon$

$\Leftrightarrow$ $\frac{1}{2}$ < $\frac{\varepsilon }{2}$

$\Leftrightarrow$ 1 < $\varepsilon$

Thus

$\varepsilon$ > 0 and $\varepsilon$ = 1 > x = $\frac{1}{2}$

Therefore x < $\varepsilon$

Q.E.D
Suppose $\varepsilon<\dfrac{1}{2}$...

What sort of consideration should one do with $\dfrac{\varepsilon}{2}$ and $x = \dfrac{1}{2}$? Then you state that $x<\varepsilon$ if and only if $\dfrac{1}{2}<\dfrac{\varepsilon}{2}$ (which is false... suppose $\varepsilon = \dfrac{3}{4}$, then $\varepsilon > x = \dfrac{1}{2} > \dfrac{3}{8} = \dfrac{\varepsilon}{2}$) then you say that is true if and only if $1<\varepsilon$ (this if and only if may be true, but is not pertinent to what you are trying to prove).

Here is a way to think about what you are trying to prove (hopefully it might even be a useful way of thinking about it):

You are given a number $\varepsilon$. All you know about it is that it is greater than zero. It could be 10 million. It could be $0.0000000000001$. You have no clue. All you know about it is that it is greater than zero. Prove that there is some number (between zero and one) that is smaller than $\varepsilon$, but still bigger than zero.

7. Re: Proving espilon > x in S

SlipEternal,

Thank you for the help, but still scratching my head. I will continue my thinking process. I clearly understand what the problem is asking me to do, except that I still confuse or unknown as how I come up with a logical reasoning.

8. Re: Proving espilon > x in S

Try this: Let $x = \min \left\{\dfrac{\varepsilon}{2},\dfrac{1}{2}\right\}$. Now show $x \in S$ and $x < \varepsilon$.

9. Re: Proving espilon > x in S

Thank you for the help SlipEternal,
I think I will get it.