Proof if $M$ is bounded, then so is its closure

Question

I want to proof that if $M\subseteq \mathbb R$ is bounded, then so is $\overline{M}$ or more precise that if $s$ is the supremum of $M$, then it is the supremum of $\overline{M}$. I came up with a proof but I am not sure if it is correct:

Let $s := \sup M$. If $M$ is closed we are done, so suppose $M$ is not closed. Suppose $s$ was not the supremum of $\overline{M}$. Then $\exists x\in \overline{M}$ with $x>s$.

By the definition of the closure there exists a sequence $(a_n)_{n\in \mathbb N_0}$ in $M$ with $\lim_{n\to \infty} a_n = x$, i.e. for all $\epsilon> 0, |a_n -x| < \epsilon$ for all big enough $n\in \mathbb N_0$.

But that means $$-\epsilon < a_n -x < \epsilon \Longleftrightarrow x < a_n + \epsilon \leq s+\epsilon$$ This is a contradiction to $x> s$.

Answer 1

Very good. To finish off, you may want to say something like: "since $x\leq s+\epsilon$ for all $\epsilon>0$, it follows that $x\leq s$, a contradiction".

Answer 2

I think you show that $s$ is an upper bound of $\overline{M}$, but you don't actually show that it's the least upper bound of $\overline{M}$.

It should, however, be straightforward to show that for any upper bound $z$ of $\overline M$, $s \le z$. This should follow from the fact that any such $z$ will also be an upper bound of $M$.

You also might want to think carefully about whether you really needed a proof by contradiction here. (See, for example, Tim Gowers' thoughts on it here.) I don't think you do, and dropping the attempt entirely but maintaining the rest of the argument still works.

Consider this reframing of your proof (with some details you've supplied omitted):

Suppose $M$ is not closed, and consider $x \in \overline{M} \cap M^c$. There must then exist a sequence $\{ a_n \} \subset M$ such that $a_n \to x$. This means that for any $\varepsilon > 0$, $x < s + \varepsilon$, which implies that $x\le s$. $x$ was arbitrary, so $s$ is an upper bound of $\overline M$.

Answer 3

I think I have a clear and simple solution using the definition of closure of M as $M\cup$ Boundary(M):

To show that it's bounded, try and latch onto this concept: suppose x is in M: i.e. let $x\in M$. Then x is in [inf(M), Sup(M)]. Now suppose x is in Boundary(M). Could we be so lucky as to have any boundary point x also be in [inf(M), Sup(M)]? That would be really easy to construct a proof out of. So suppose there was a boundary point b outside of this interval:

We'd have either $b<$inf(M) or $sup(M)<b$. In either case, there'd be a neighborhood N of points around b so that $N\cap M= \varnothing$. But since b is a boundary point of M and N is a neighborhood of b, there must be some point m in M that is also in V which is impossible because because $M\cap$[inf(M),sup(M)=$\varnothing$.

The formal proof would go something like:

Let x be in cl(M).

Then x is in $M\cup bd(M)$.

Since x is in M or x is in Bd(M), then....

and you'd proceed to consider each case in the concept above...ending with the threat of contradiction in case two to conclude that regardless of which case it is, x is in [inf(M),sup(M)] and thus cl(M) is bounded.

Let me know if you have any questions.

Adam V. Nease