Maximal Elements in a Bounded Set

[ertagon2]

Could someone please check these questions? Please correct them if necessary, with an explanation if you could.
View attachment 7937

 

[castor28]

Hi ertagon2,

Everything looks OK, except 5b. Think about the set $\displaystyle\left\{1-\frac{1}{n+1}\:\bigg|\: n\in\mathbb{N}\right\}$.

 

[Evgeny.Makarov]

I agree with everything except 5.2. Consider an open interval.

 

[ertagon2]

Hi ertagon2,

Everything looks OK, except 5b. Think about the set $\displaystyle\left\{1-\frac{1}{n+1}\:\bigg|\: n\in\mathbb{N}\right\}$.

I don't think I understand. Can you elaborate?

 

[castor28]

I don't think I understand. Can you elaborate?

Hi ertagon2,

This is the set $\displaystyle S=\left\{0,\frac12,\frac23,\frac34,\ldots\right\}\subset\mathbb{Q}$. This set is bounded above (by $1$). In fact, $1$ is the least upper bound of $S$, but it is not an element of $S$.

No element of $S$ can be maximal, because, for each element $\left(1 - \dfrac{1}{n+1}\right)\in S$, $\left(1 - \dfrac{1}{n+2}\right)$ is greater and also an element of $S$.