I don't think that it is necessary for P to be an interior point of S to prove it.

First, compactness: if S' has an open cover, then the inversely scaled cover is an open cover of S, hence has a finite subcover for S, and the scaled version of this is a finite subcover of the original open cover of S'.

Second, convexity: the inverse scaling of two points of S', say, are two points of S. If S is convex, then the entire line segment between these inversely scaled points must be in S, and therefore the scaled version of that line segment must be in S'. So to really show this in detail, you need to show that the scaled segment between and is identical to the line segment between and .