In Guillemin and Pollack, the Stability Theorem states that the following classes of smooth maps of a compact manifold X into a manifold Y are stable classes: local diffeomorphisms, immersions, submersions, maps transversal to any specified submanifold ZY , embeddings, and diffeomorphisms. A property is stable provided that whenever f0:XY possesses the property and ft:XY is a homotopy of f0 , then, for some >0 , each ft with t< also possesses this property. The proof of the stability theorem is in Guillemin and Pollack, but I want a complete proof of the following: Suppose fs:XY is a smooth family of mappings if the map F:XSY , defined by F(x,s)=fs(x), is smooth. Check that the Stability Theorem generalizes immediately to the following: if f0 belongs to any of the classes listed, then there exists >0 such that fs belongs to the same class if s0s< .