Is the set $A=\{\frac1n,n\in\mathbb N\}$ bounded? I am studying the following set
$$
A=\left\{\frac{1}{n},n\in \mathbb N\right\}
$$
Where $\mathbb N$ begins from $1$ not $0$. I know that the upper bound of this series is $1$ but what is the lower pound? Is it $0$ but $0$ doesn't exist in $N$ because $N=(0,1]$.
Does the lower/upper bound have to exist in the set. What I mean is can I pick for example $2$ to be the upper bound or $-2$ to be the lower bound?
Finally, is there an examples of (1) a set that is bounded from one end only and (2) an example of a set that is not bound from both ends?
thanks in advance.
 A: Yes, the greatest lower bound of $A$ is $0$. First, it’s clear that $0<\frac1n$ for each $n\in\Bbb Z^+$, so $0$ is a lower bound for $A$. Secondly, if $x>0$, then there is an $n\in\Bbb Z^+$ such that $\frac1n<x$, so $x$ is not a lower bound for $A$. Thus, $0$ must be the largest lower bound for $A$.
Any real number smaller than $0$ is also a lower bound for $A$, so $-2$ is indeed a lower bound for $A$. Similarly, any real number larger than $1$ is an upper bound for $A$, so $2$ is an upper bound for $A$.
The set $\Bbb Z^+$ is bounded below (by any $x\le 1$) but not above, and $\Bbb Z$ is not bounded at either end. You can probably easily come up with other examples.
A: I am studying the following set
$$ A=\left\{\frac{1}{n},\;n\in \mathbb N\right\}$$
where $\mathbb N$ begins from $1$ not $0$. I know that the upper bound of this set is $1$ but what is the lower bound?
A upper bound is $1$ and a lower bound is $0$, but there others.
Does the lower/upper bound have to exist in the set?
No. ("greatest/least element" have to exist in the set, but  "lower/upper bounds" and "supremum/infimum" don't.)
What I mean is can I pick for example $2$ to be the upper bound or $-2$ to be the lower bound?
Yes.
Finally, is there an examples of (1) a set that is bounded from one end only and (2) an example of a set that is not bound from both ends?
Yes:
$$\{n,\;n\in\mathbb{N}\}\tag{1}$$
$$\{-n,\;n\in\mathbb{N}\}\cup\{n,\;n\in\mathbb{N}\}\tag{2}$$
