How to prove that there exists an $x\in\mathbb R$ such that $x$ is smaller than or equal to all given elements of any subset of real numbers? Let $Z$ be a subset of the real numbers.
• Jesse likes $Z$ if and only if $∃a ∈ \mathbb{R}$ such that $∀x ∈ Z, a ≤ x$.

Does Jesse like every subset of the real numbers? 
I answered yes, because since $Z$ is a subset of $\mathbb{R}$, there has to be a real number that is equal to or less than any number in $Z$. 
However, how do I prove this mathematically? If it's not true, where did I go wrong and where would I start to prove that it is not?
Thank you very much.
 A: The number $a$ that Jesse wants to exist is a lower bound of the set $Z$.
If a set of real numbers has only a finite number of members, it must have
a greatest element and also a least element, therefore it has an upper bound
and a lower bound.
But an infinite set of real numbers might not have an upper bound.
It might not have a lower bound either.
Remember that the every set is a subset of itself.
What number in $\mathbb R$ is less than or equal to every number in $\mathbb R$?
A: This is not true, for example take the trivial case 
$$
Z_1:=\mathbb{R}\subseteq\mathbb{R}
$$
since the set itself is a subset of itsself. Another example is the open intervall with $b\in\mathbb{R}$
$$
Z_2:=(-\infty,b)
$$
where you can't find any $a\in\mathbb{R}$ such that all elements in $Z_2$ are greater equal $a$, since $Z_2$ has no bound from below.
A: You don't have to prove that it's not.  You just have to give a counter example.
The statement $∃a ∈ \mathbb{R}$ such that $∀x ∈ Z, a ≤ x$, is the definition for Z being "bounded below".  Not all subsets are bounded below.  $\mathbb{R}$ is not bounded below. (It is not the case that there is an x equal are smaller than all real numbers.)  The integers are not bounded below. (It is not the case that there is a number that is as small or smaller than all integers.)  
An example of an infinite set that is bound below is the positive real numbers.  0 is smaller than all positive real numbers.  Se Jesse loves the set of positive real numbers.
