Prove that if X is well-ordered, there is no strictly decreasing sequence of elements in X.

Question

Im not sure how to go about this. I know this isn't correct but so far this is all I can come up with. Assume that $(x_i)$ is a strictly decreasing sequence in $N$. Then we can say that $x_i$ such that $i \in N$ is a nonempty subset that has no smallest element by definition. We know that for $x_1, x_2 \in N$ that $[x_1 < x_2 ]\ \Rightarrow\ [x_2 \preceq x_1\ \wedge x_1 \neq x_2]. $ The well ordering principle requires that there is a smallest element, therefore this is a contradiction and we have shown that there is no strictly decreasing infinite sequence in $N$.

Answer 1

Consider the set of such elements. What is the minimum of such set?