Are there subsets of $\mathbb{Q}$ with a finite number of upper-bounds?

Question

Consider the poset $(\mathbb{Q}, \leq)$. Is there a subset of $\mathbb{Q}$ with a finite number of elements that are upper-bounds?

I tried to prove this as follows:

Suppose $K \subset \mathbb{Q}$, if $x \in K$ then $x+1 \in \mathbb{Q}$ and $x+2 \in \mathbb{Q}$ and so goes. This implies that for any subset of $\mathbb{Q}$ there is an infinite number of elements that are upper-bounds.

Answer 1

Just take $\mathbb{Q}$. It has no upper bounds. More generally, you can take any subset of $\mathbb{Q}$ that is unbounded above (there are many).