I am working on the following problem: If a set of integers $S \subset \mathbb{N}$ has a supremum, show that $\sup S \in S$.
My approach is as follows:
Let $s_0 = \sup S$ and suppose $s_0 \notin S$. It is a fact that for every $\epsilon > 0$
there exists $a \in S$ such that
$$
s_0 – \epsilon < a \leq s_0.
$$
However, since $s_0 \notin S$ and $a \in S$ we in fact have
$$
s_0 – \epsilon < a < s_0
$$
But since $\epsilon > 0$ it follows that
$$
s_0 – \epsilon < a < s_0 + \epsilon \implies -\epsilon < a – s_0 < \epsilon \implies |a – s_0| < \epsilon
$$
Since this is true for every $\epsilon > 0$, $a – s_0 = 0 \implies a = s_0$, a contradiction. Therefore, $s_0 \in S$
My question then is whether this argument is valid. I believe it is correct but it is significantly different from the author's solution so I would like a second opinion.
Best Answer
Note that nowhere in your argument did you use the fact that $S$ is a set of integers. If the argument were valid, why is it not valid for sets of real numbers?
The error is that the witness $a$ actually depends on $\epsilon$. You have no warrant for assuming a priori that you can always pick the same $a$, regardless of what $\epsilon$ is, so the implication that leads to $a-s_0=0$ is invalid.
To write this properly, you would have to say:
And now it should be clear why you cannot go from this to "Therefore, $a(\epsilon)-s_0 = 0$."
Again: the tip-off should have been that you never used the fact that every element of $S$ is an integer.
Now, to fix this, we can actually proceed directly: let $s_0$ be the supremum of $S$. Then, since $s_0$ is the supremum, for every $n\gt 1$ there exists $a_n\in S$ such that $$s_0-\frac{1}{n} \lt a_n \leq s_0.$$ Now notice that for all $n$ and $m$, then $$|a_n-a_m|\leq |a_n-s_0|+|s_0-a_m| \lt \frac{1}{n}+\frac{1}{m}\leq \frac{1}{2}+\frac{1}{2} = 1.$$ So $|a_n-a_m|\lt 1$. But $a_n$ and $a_m$ are both integers. (Aha! We knew we had to use that sometime...) Since they are both integers, and they differ by less than $1$, then $a_n=a_m$.
Therefore, there is a single $a\in S$ such that $s_0-\frac{1}{n}\lt a\leq s_0$ for all $n\gt 1$.
Now you can finish the argument as you did, to conclude that $|a-s_0|=0$, so $a=s_0$, hence $s_0\in S$.