Can we put a sphere with a bigger radius inside of a smaller one so that it would be fully contained? The answer is yes, but we have to select the space we are working in carefully. I am going to provide two examples here:
First example: metric space
Suppose we are in a metric space with euclidean metric. The space is a sphere itself, with radius . Let us denote a closed sphere of radius with center at 0 as . This sphere is a subset of .
What is a sphere? By definition it is a point of a space and a set of such points that are no further than the sphere’s radius. What if we try to define a sphere of radius inside the space we are working in?
Why does this sphere look as an intersection of two? We are, by definition, effectively selecting a subset of points of space whose distance to is less or equal to . Any point of lying outside of is out of reach.
As long as , we can fit a bigger sphere inside of a smaller one. Once we cross the border, even if we pick a center on the border of the , the longest distance between two points inside of can not surpass . Thus, all points will be contained inside of .
We can also pick up a less traditional kind of space, for example, a graph. Let us label its edges with numbers and take the shortest path as a metric.
It is easy to see that , for: