For every ε>0 there is an a∈A such that a<z+ε

**alice8675309** · January 27th 2011, 07:46 PM

So there are a few questions that go along with this but I've hopefully worked them out correctly so I am only posting the ones I definitely have no clue about.

Let A be a bounded set of reals and z a real number such that the following holds: For every ε>0 there is an a∈A such that a<z+ε (if it is true, prove it. If not, then show a counterexample.

(1) inf A $\leq$ z

I said it was TRUE but I don't know how I would prove it.

(2) inf A $\geq$ z

This one I have no clue about. I'm assuming its FALSE because I said the above statement is true.

(3) sup A $\leq$ z

I said True by trichotomy..but I know that isn't a proof so how would I prove this one as well.

Thanks!

**FernandoRevilla** · January 28th 2011, 01:03 AM

Denote $m=\inf A\;,\;M=\sup A$

(1) True. By contradiction:

If $m>z$ choose $\epsilon=m-z$ .

(2) False. Choose $A=[0,1)\;,\;z=1$ .

(3) False. Choose $A=[0,1)\;,\;z=1/2$ .

Fernando Revilla

**alice8675309** · January 30th 2011, 04:38 AM

Thanks! Now if the same conditions hold For every $\epsilon$ >0 there is an a $\in$ A such that a<z+ $\epsilon$ : what about..

1. A has an element a $\geq$ z
I said false: [-1,1]=A , z=1 (or would it be the open set (-1,1)?)

2. Inf A=z
I said false: (0,1)=A, z=1

3. Sup A=z
I said false: [-2,-1] z=0 ( would this counterexample also work for sup A $\geq$ z as well?)

**Plato** · January 30th 2011, 04:50 AM

Originally Posted by alice8675309

Thanks! Now if the same conditions hold For every $\epsilon$ >0 there is an a $\in$ A such that a<z+ $\epsilon$ : what about..
1. A has an element a $\geq$ z
I said false: [-1,1]=A , z=1 (or would it be the open set (-1,1)?)

Did you read reply #2?
The above, 1), is true. In #2 is a proof of that.

**alice8675309** · January 30th 2011, 05:03 AM

Originally Posted by Plato

Did you read reply #2?
The above, 1), is true. In #2 is a proof of that.

Sorry, I did read reply #2 but I was honestly checking the answers I had. I think I'll have to draw a picture to connect everything.

**alice8675309** · January 30th 2011, 05:23 AM

Originally Posted by Plato

Did you read reply #2?
The above, 1), is true. In #2 is a proof of that.

Now looking at my paper, the reason I had posted the question A has an element a $\geq$ z and said it was false was because I had another question with the same conditions stated in the first post that said:

A has an element a $\leq$ z and I said True because if a=z then a<z+ $\epsilon$ holds for all $\epsilon$ >0 and if a <z then a<z+ $^epsilon$ holds for all $\epsilon$ >0.

Do I have them backwards? Which one is true and which one is false and what would be a counterexample for the false one?

**Plato** · January 30th 2011, 05:39 AM

Given that $\left( {\forall \varepsilon > 0} \right)\left( {\exists a \in A} \right)\left[ {z \leqslant a < z + \varepsilon } \right]$ .

If $m=\text{inf}(A)$ suppose that it is true that $z<m$ . Then let $\varepsilon =(m-z)>0$ .

So $\left( {\exists a \in A} \right)\left[ {z \leqslant a < z + (m - z)} \right]$ .
But that says $a<m$ . What is wrong with that?

Thus we have that $m\le z.$

**alice8675309** · January 30th 2011, 05:51 AM

Originally Posted by Plato

Given that $\left( {\forall \varepsilon > 0} \right)\left( {\exists a \in A} \right)\left[ {z \leqslant a < z + \varepsilon } \right]$ .

If $m=\text{inf}(A)$ suppose that it is true that $z<m$ . Then let $\varepsilon =(m-z)>0$ .

So $\left( {\exists a \in A} \right)\left[ {z \leqslant a < z + (m - z)} \right]$ .
But that says $a<m$ . What is wrong with that?

Thus we have that $m\le z.$

Ohhh I see it now! So for the one that states A has an element a $\leq$ z does this work as a counterexample? A=[0,1) z=1?

**Plato** · January 30th 2011, 06:07 AM

Originally Posted by alice8675309

Let A be a bounded set of reals and z a real number such that the following holds: For every ε>0 there is an a∈A such that a<z+ε (if it is true, prove it. If not, then show a counterexample. (1) inf A $\leq$ z
(2) inf A $\geq$ z
(3) sup A $\leq$ z

Above I have quoted the OP.
I answered (1), proved it is true.
Both (2) & (3) are false.
To show that we need to find an example of both the set A and the value z.
Let $A=[0,1]$ then $0=\inf(A)~\&~1=\sup(A)$ .
If we choose $z=\frac{1}{2}$ both are false.

**alice8675309** · January 30th 2011, 06:17 AM

Originally Posted by Plato

Above I have quoted the OP.
I answered (1), proved it is true.
Both (2) & (3) are false.
To show that we need to find an example of both the set A and the value z.
Let $A=[0,1]$ then $0=\inf(A)~\&~1=\sup(A)$ .
If we choose $z=\frac{1}{2}$ both are false.

Thanks so much! I'm sorry if it got confusing because I did post three additional questions in reply #3 which I guess I should have started a new thread for. However, are the counterexamples in reply #3 for 2 and 3 by any chance correct? Again thanks you've been a big help! This topic isn't my strongest

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