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.
[Math] 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
real numbersreal-analysis
Best Answer
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.