Equivalent Definition of Well-Ordered Set

Question

On Wikipedia <https://en.wikipedia.org/wiki/Well-order>, it states the following.

If a set is totally ordered, then the following are equivalent to each other:

1. The set is well ordered. That is, every nonempty subset has a least element.
2. Transfinite induction works for the entire ordered set.
3. Every strictly decreasing sequence of elements of the set must terminate after only finitely many steps (assuming the axiom of dependent choice).
4. Every subordering is isomorphic to an initial segment.

Question: How does one show 4 to and from the rest? (I welcome simply posting a reference to an outline proof)

Remark: I understand that 1--3 are different ways of saying that the strict ordering $<$ is well-founded.

Edit: I retract my question about showing 4 from the rest. It has to do with order-preserving injections. This is related to a previous question of mine: For each ordinal $\alpha$, $\alpha\le \aleph_{\alpha}$. What remains is from 4 what is an argument for the rest?

Answer 1

Suppose you have a totally ordered set $A$ satisfying (4).

If it is empty, then it clearly satisfies (1).

Otherwise it has an element $a$, and $\{a\}$ is isomorphic to a initial segment. This means that $A$ has a first element.

Now every non-empty subset $B\subseteq A$ induces a subordering, which must be isomorphic to an initial segment. This isomorphism maps the first element of $A$ to a first element of $B$.