I've found in an article the following statement:
Consider a compact metric set covered by $n$ balls of radius less than $\epsilon/3$. Now consider the balls with the same center and doubled radius (implying that all $r$ are $<2\epsilon/3$). The Lebesgue number of this new cover is $\geq \epsilon/4$.
Note that this is subtler than it could appear: we are not saying that the Lebesgue number of any cover of radius $2\epsilon/3$ is more or equal than $\epsilon/ 4$: this is true only if the cover comes from an $\epsilon/3$- cover by rescaling; note that rescaling by $1/2$ a cover we don't necessarily obtain a cover of the space.
Now can someone help me proving the given statement?
Thank you in advance.