Covering of closed unit ball with closed balls.

Question

Notations and definitions

Let $E$ be a finite dimensional vector space with norm $||\;||$.

Let $B$ denote the closed unit ball in $E$ and $B_r[a]$ the closed ball centered at $a$ with radius $r$.

I will say $B$ has a finite cover with closed balls of radius $r>0$, if for a given $r >0$, $$\exists N\in \mathbb N, \exists (a_1,\ldots,a_N)\in E^N, B \subset \cup_{i=1}^N B_r[a_i]$$

With the help of sequential compactness, I was able to prove the existence for any $r>0$ of a finite cover of $B$ with closed balls of radius $r$.

Now let $m(r) \in \mathbb N$ be the least number of closed balls of radius $r$ needed to cover $B$.

Question

I am asked to prove that the well-defined function $$\begin{array}{ccccc} m & : & \mathbb R_+ & \to & \mathbb N \\ & & r & \mapsto & m(r) \\ \end{array}$$

is decreasing.

This result is very intuitive : you need less bigger balls to cover $B$, but I'm unable to prove it...

How can I link two covers of $B$ with different radii? Should I go for contradiction ?

Answer 1

This is obvious. Note that $m(r)$ is not only an infimum, but a minimum. This means that you can cover $B$ with $m:=m(r)$ suitably placed balls of radius $r$. When $r'>r$ then $m$ balls of radius $r'$ with the same centers will obviously cover $B$, but there might be a better covering with balls of radius $r'$. So $m(r')\leq m(r)$. (You cannot hope for strict monotonicity.)