Homework Statement

Consider the sets below. For each one, decide whether the set is bounded above. If it is, give the supremum in ##\mathbb{R}##. Then decide whether or not the set is bounded below. If it is, give the infimum. Finally, decide whether or not the supremum is a maximum, and whether to not the infimum is a minimum.

c) the natural numbers ##\mathbb{N}##

Homework Equations

In my class ##\mathbb{N} = {0, 1, 2, ... }##

The Attempt at a Solution

Proof: ##\mathbb{N}## is not bounded above. We will show this with a contradiction. Suppose M is an upper bound on ##\mathbb{N}##. Then (##\lceil M \rceil + 1) \space \epsilon \space \mathbb{N}## and ##\lceil M \rceil + 1 > M##, a contradiction. Therefore ##\mathbb{N}## does not have an upper bound and is not bounded above.

We will now show ##\mathbb{N}## is bounded below. Let ##U = 0## and let ##x \space \epsilon \space \mathbb{N}##. Then ##U = 0 \le x##. Therefore ##0## is a lower bound on ##\mathbb{N}##.

In order to show inf##\mathbb{N} = 0## we must show 0 is the greatest lower bound. We proceed by contradiction. Suppose ##m## is a lower bound on ##\mathbb{N}## such that ##m > 0##. But ##0 \space \epsilon \space \mathbb{N}## so m is not a lower bound, a contradiction. We conclude that there does not exist a lower bound ##m## such that ##m > 0## and so inf##\mathbb{N} = 0##.

In order to show min##\mathbb{N} = 0##. we must show 0 is a lower bound on ##\mathbb{N}## and ##0 \space \epsilon \space \mathbb{N}##. We've already shown 0 is a lower bound on ##\mathbb{N}## and ##0 \space \epsilon \space \mathbb{N}## is a true statement. Therefore min##\mathbb{N} = 0##. []

My question: Is this proof clear and easy to follow?