The goal is to trace a line with your pencil across each edge on the box only once without crossing the vertices or picking your pencil up.

Note that this "box" contains 16 unique "edges".

Prove why this is an impossible task regardless of where you first place your pencil.

Michael, what I'm trying to say and certainly I am being misunderstood (excuse if this is not correct in english), is the following : if you "quote" the results of Euler in his study involving networks, results obtained trough a extensive proof, and apply these results to a particular network, you are not proving. 

One thing is "prove that it is impossible", other is "show that it is impossible".

It seems only a matter of semantics, but they are quite different.

And I apologyze to Eric if I didn't make myself clear.