Tessellate S.T.E.M.S - Computer Science - College - Set 1 - Problem 3

Consider the set $\mathbb{N}^*$ of finite sequences of natural numbers. Let $P = \{ (x,y): x \text{ is a prefix sequence of } y \}$ . Let $P$ induce the relation $<_P$ , where $x <_P y$ if $(x,y)$ are in $P$ . Which of the following are true?

1. $\mathbb{N}^*$ is countable.
2. $<_P$ is a well quasi-order .
3. $<_P$ is a total order.
4. Every non-empty subset of $\mathbb{N}^*$ has a least upper bound.
5. Every non-empty subset of $\mathbb{N}^*$ has a greatest lower bound.

This problem is a part of Tessellate S.T.E.M.S.