Simple and intuitive example for Zorns Lemma Do you know any example that demonstrated Zorn's Lemma simple and intuitive? I know some applications of it, but in these proof I find the application of Zorn Lemma not very intuitive.

It is not intuitive. It's not without reason that this joke is well known:

"The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?" — Jerry Bona

I defer to K. Conrad's "blurb" about Zorn's lemma: http://www.math.uconn.edu/~kconrad/blurbs/zorn1.pdf: 

Zorn's lemma provides no mechanism to find a maximal element whose existence it asserts. It also says nothing about how many maximal elements there are. Usually [...] there are many maximal elements.

In particular note that the upper bound created during a proof using Zorn's Lemma does not have to be a maximal element. As K. Conrad remarks in Example 2.3, the ideals $\{6\mathbb Z, 12\mathbb Z, 24\mathbb Z\}$ in $\mathbb Z$ have $6\mathbb Z$ as an upper bound (with the partial ordering just being set inclusion), but $6\mathbb Z$ is clearly not maximal among proper ideals in $\mathbb Z$.
