Another AC problem

Question

I am tripping over these examples I have to show that any chain C can be expanded to a maximal chain M, using AC

I know the def. of maximal chain So say C is chain it’s elements C={x1,x2...,xn} with a cardinal number |C|=n Define another set M, with associated elements, |M|=m Def of maximal element So if M subset C. Bc iff m < n By AC f:C->M take some x€C { by def of AC take elements of each subset & put in f as long m < n}

Answer 1

Let S be a partially ordered set.
Let K be the set of all chains within S ordered by subset.
Let C be a chain within K.
Show $\cup$C is in K and is an upper bound of C.
Thus by Zorn's lemma, K has a maximal element,
a maximal chain within S.