Questions about manifolds 
This is a question from spivak and "proper" means the inverse image of any compact set of N is still compact. However, I can not find a suitable compact subset of N to use this property.
 A: Look at the image of $f.$ If you show that it is open and closed, then it is a connected component of $N,$ and since $N$ is connected, it is the whole thing. The properness comes in showing that the image is closed: if $p_1, \dots, p_n, \dots$ converge to $p$ in $N,$ then, since the preimage of a closure of a neighborhood of $p$ is compact (and non-empty, since it contains all but a finite number of the convergents), then for any sequence of $q_1, \dots, q_n, \dots$ such that $f(q_i) = p_i$ you can pick a convergent subsequence, and the image of the limit is $p,$ by continuity. I leave openness to you...
By Popular Demand Some comments about the other conditions: They are needed to show openness. First, the orientation condition is necessary (think of projecting the round sphere on its first coordinate). Second, to respond to Georges' concern, you can't have just one regular point, through the magic of Sard's theorem. Either all the points are critical (obviously possible), or the set of critical points has hausdroff dimension at most $n-1.$ The points where the nullity is two or more can be ignored (since they don't separate), at a point where the nullity is $1,$ there is potentially a separating set of critical points, but the only way the map can't be extended past this set is if the map changes orientation there.
Now, if this is an exercise, I have no idea if Spivak had introduced Sard's theorem at this point, but I don't really see how to avoid it.
