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**alice8675309** This is a list of 10 properties in which they are either true or false in the context of the following. Some I've attempted a solution, which I hope someone will be able to confirm my solution as being right or if it is wrong let me know where I went wrong. And there are some that I know are true or false I just don't know how to start proving them so I was hoping someone could lead me in the right direction on how to prove or provide a counterexample.

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. A$\displaystyle \neq$ ∅

My answer: TRUE. If A=∅ there are no elements in a so there does not exist any a$\displaystyle \in$A such that a<z+ε for every ε>0.

2. z$\displaystyle \in$A

my answer: False. Counterexample: (0,1)=A z=1

3. A has an element a$\displaystyle \leq$z

my answer: TRUE. If a=z then a<z+ε holds for all ε>0. If a<z then a<z+ε holds for all ε>0

4. A has an element a$\displaystyle \geq$z

my answer: FALSE. Counterexample [-1,1]=A z=1 ?

5. inf A=z

my answer: FALSE. Counterexample (0,1)=A z=1

6. inf A $\displaystyle \leq$z

my answer: TRUE (not sure how to prove this one)

7. infA $\displaystyle \geq$z

i'm not sure at all about this one

8. sup A=z

my answer: FALSE. Counterexample: [-2,-1] z=0

9. sup A $\displaystyle \leq$z

My answer: TRUE. Based on trichotomy? I don't know how to prove this one.

10. sup A$\displaystyle \geq$ z

my answer: FALSE. Counterexample: [-2,-1] z=0

I know its quite a bit but i'm just hoping that someone can tell me if my answers are right or wrong, if my proofs are good and if not explain how to prove it or if one is lacking a proof give some kind of direction and also if my counterexamples are good. Thanks!