I need some assistance with the following problems. I've come up with some ideas but may need some clarification.
1) Let $S$ be a well-ordered set.
a) Need to show $S$ has a first element.
Solution: We know that any subset of $S$ has a first element, and since $S$ $\subseteq$ $S$, then clearly $S$ must have its own first element.
b) Need to show that $S$ is linearly ordered.
c) Show that $S$ is order-complete; that is, every subset that is bounded above has a suprenum.
My intuition for this is to relate this to the Completeness Axiom of $\mathbb R$.
d) Let $A$ be a non-empty subset of $S$, and show $A$ is well-ordered using the same ordering.
2) Let $f:$ $A \rightarrow$ $B$ be a surjective function; show $\exists$ an injective function $g:$ $B \rightarrow$ $A$. (This would imply $|A| \geq |B|$).