To take the easy part first, you can soon check that N' is a unit vector. In fact,

(because and the other two terms cancel).

The normal vector N can be characterised as the unique unit vector with the property that , where the limit is taken along any path in the surface S. We want to show that the vector N' has this property for the surface F(S). So we need to show that . The numerator of that fraction is which with a bit of juggling you can rearrange as

It follows that As , both of those terms go to zero. That shows that N' is normal to F(S) at F(p).