Suppose $\emptyset \neq A \subset \mathbb{R} $. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$
This is my attempt:
$A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 \leq x_i < 2$ so clearly $2$ is an upper bound.
To show it is the ${\it least}$ upper bound, suppose that $2 \neq \sup A$, that is there exists a number $M < 2$ for some real $M$ qualifying as $\sup A$. Certainly this $M \in [0,2) $ so $ M > 0 \Rightarrow 2 -M > 0$.
By the Archimedean Principle, for all real numbers $r > 0\,\, \exists\,\, n \in \mathbb{N}$ such that $0 < \frac{1}{n} < r $. By the Approximation Property of Suprema, there exists $a \in [0,2)$ such that $\sup A – \epsilon < a \leq \sup A$, where $\epsilon > 0$.
Suppose $\sup A = M < 2$. Then the above gives $M – \epsilon < a < 2\,\,\,\,\forall \epsilon > 0$. Also, by Archimedean, we have $0 < \frac{1}{n} < 2-M$, so choose $\epsilon = 2-M$. Then $M – (2-M) < a < 2 \,\Rightarrow 2(M-1) < a < 2$
We can assume $M – 1>0$ and so $2(M-1) > 2$ This results in a contradiction in the previous inequality. Hence $M < 2$ cannot be the supremum.
I realise there is probably a simpler way, but is what I have written all good?
Best Answer
I think there's a way more simple and intuitive proof.
First, as you observed, it is obvious that 2 is an upper bound. Now, to prove it is the supremum. Assume that $M$ is the supremum and $M<2$.
Of course, $M>2$ is trivially impossible since $2$ is an upper bound as well and thus any $M$ bigger than $2$ cannot be a supremum.
Now, let $x=\frac{2-M}{2}+M$.
As you can obviously see $M<x$, so all that is left is to show that $x<2$.
But now, assume it is not, that is $x\geq2$.
Then, $\frac{2-M}{2}+M\geq2$. Multiplying both sides by $2$, we get
$2-M+2M\geq4$, that is $2+M\geq4$. But $M<2$, so $2+M<4$. Thus, by reductio ad absurdum, $x<2$. This shows that $M<2$ cannot be the supremum.
QED.
Now, as for the simplicity of this proof, I have written a lot for clarity and in case you are a beginner on this subject. This can be summarized in $2$ lines, but this is for clarity. I hope this helps, and you must soon learn to find the shortest and more intuitive way. Good luck.