There are two cases: either the graph contains a cycle or it does not. If the graph contains a cycle, then pick an edge that is part of this cycle, and remove it. This will not disconnect the graph, and will decrease the number of faces by 1 (since the edge was bordering two distinct faces). So by the inductive hypothesis we will have \(v - k + f-1 = 2\text{.}\) Adding the edge back will give \(v - (k+1) + f = 2\) as needed.
in-context