I'm having some trouble with the definition of "Bounded Set". I have a pretty good idea of what "Limited" means: a Set with a Upper and a Lower bounds. Now i have a quiz in which I must choose the right answer and I have absolutely no idea what to chose:
With A ⊆ R and M ∈ R+, A is Limited if:
(a) ∀M ∈ R+ : ∃a ∈ A : |a| > M
(b) ∃a ∈ A : |a| > M, ∀M ∈ R+ :
(c) ∃M ∈ R+ : |a| ≥ M, ∀a ∈ A
(d) ∃M ∈ R+ : ∃a ∈ A : |a| > M
(e) ∀M ∈ R+ : |a| ≥ M, ∀a ∈ A
In the same way:
With A ⊆ R and M ∈ R+, A is Unlimited if:
(a) ∀M ∈ R+ : ∃a ∈ A : |a| > M
(b) ∃M ∈ R : ∃a ∈ A :|a| > M
(c) ∀a ∈ A : ∃M ∈ R+ :|a| ≥ M
(d) ∃M ∈ R+ : |a| ≥ M, ∀a ∈ A
(e) ∀M ∈ R+ : |a| ≥ M, ∀a ∈ A
Can you chose the right answer? ( I have the solutions of course but i want a clear explanation of what an limited and unlimited set is). Thanks
Edit: the right answers: (c) and (a)
For the first one, you could ask your self which of the statements are true for $A= \mathbb{R}$ itself (assuming here that $R$ is the real numbers.
For example the first statement if true for the real numbers since indeed for any real number $M$ ($\forall M$ ) you can find another number $a$ ($\exists a$) such that the absolute value of $a$ is greater than $M$ ($\lvert a \lvert > M$). This shows that the first statement does not say that $A$ is bounded/limited since the example with $A=\mathbb{R}$ is unbounded.
Try to analyse each statement like that and remember that the empty set is bounded.
And oops, I think I just gave away the answer to the second problem.
Edit: You indicate in your question that you believe that (c) is the correct answer for the first problem. Try to consider the set $A = [2,\infty)$. Then $A$ is unbounded, but does not the statement hold with $M = 1$?