Show that no matter in what way we embed a planar graph, we always get the same number of faces.
This is trivial for connected graphs, because Euler's formula applies and shows that $f = 2 - n + e$, and no matter how we embed the graph, the number of vertices and edges is already fixed.
I think that for disconnected graphs the formula helps as well: say we have $m$ components. Then we add one edge for each component so $m-1$ edges to connect them. The formula now applies, and we have added $m-1$ edges so $f = 2 - n + e + m-1$. The outer face is the same because we do not create new faces, so when we remove the added edges we get $f$. Maybe I am being too obnoxious about this - is there an easier way? Also is my idea correct?