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 $r$. Let us denote a closed sphere of radius $r$ with center at 0 as $D_r(0)$. This sphere is a subset of $R^n$.

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 $R > r$ 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 $D_r(0)$ whose distance to $B$ is less or equal to $r$. Any point of $R^n$ lying outside of $D_r(0)$ is out of reach.

As long as $% $, we can fit a bigger sphere inside of a smaller one. Once we cross the $2r$ border, even if we pick a center $B$ on the border of the $D_r(0)$, the longest distance between two points inside of $D_r(0)$ can not surpass $2r$. Thus, all points will be contained inside of $D_R(B)$.

# Second example

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 $D_{1.5}(D) \subset D_{1}(C)$, for: