I'm trying to prove the following claim: for any $A \subseteq R^m$, scalar $c>0$ and vector $v_0 \in R^m$ we have: $\forall r>0 , $ $ N(r,\{ca + v_0 : a\in A\}) \leq N(cr,A) $
where $N(r,A)$ is the cardinality of the smallest set that $r-covers$ $A$
I tried to prove that if $A'$ is a $cr-cover$ of $A$ then it must be a $r-cover$ of the set $ \{ca + v_0 : a\in A\} $, but I couldn't prove it.
Any other ideas? thanks.
The statement you tried to prove is wrong, but you are on the right track. Try proving instead that if $A'$ is a $cr$-cover of $A$, then $\{ca'+v_0 : a' \in A\}$ is an $r$-cover of $\{ca+v_0 : a \in A\}$.