This is demonstrated by triangulating the surface, and showing that the triangulation is not orientable.
Thanks for your answer! Unfortunately i do not know what a triangulation is. But this should work:
Let us pick a continuous curve c on the strip. If we pick a normal vector at c(0) it will point in the other direction at c(2pi). This is geometrically obvious and we would have a contradiction. But how can i show that formally?