I was studying differentiable manifolds and smooth manifolds. While reading on the Wikipedia website about them, I came across one statement that I have no idea why is it. This statement is
Every $C^k$-structure is uniquely smoothable to a $C^∞$-structure, i.e., for every $C^k$-structure with $k>0$, there is a unique $C^k$-equivalent $C^∞$-structure.
I think that this is a beautiful result, because it implies that there is no distinct notion of a $C^k$-manifold ($k>0$, if $k=0$ then not necessarily hold) and a $C^∞$-manifold, which I think will justify the 'interchangeability' of the word 'smooth' and 'differentiable' when we talk about manifold.
Anyone has any idea why this result holds? I also read that it is due to a theorem by Whitney, but I have no idea which theorem is it. If anyone can recommend me some useful material, I will definitely greatly appreciate.
Thanks in advance!