a SolutionWe will show that each finite set A has a minimal

Solution

We will show that each finite set A has a minimal element.
The existence of a maximal element then follows from duality.
For all n in N>0, Let P(n) be the proposition:
Let A have exactly one element x.
Since x is not less than x. So it follows that x is minimal.
Suppose that every subset of A with n elements has a minimal element.
Let A have n+1 elements.
Then :
A = A0U{x}
Where A0 has n elements and x in A-A0
Then A0 has a minimal element m0.
If m0 is not a minimal element of A,then : x < m0
thus x is a minimal element of A.
Thus either m0 or x is a minimal element of A.
The result follows by the principle of Mathematical Induction.