Why is the doubling dimension of any net of a metric space at most half of that of the metric space?

Definition Doubling dimension ($\dim_D(M)=k$): A Metric space $M=(V,d)$ has doubling dimension at most $k$ if for any $x\in V$ & $r>0$, $B(x,r)\subseteq\bigcup^{2^k}_{i=1}B(x_i,r/2)$.With this definition, a paper that i have read stated without proof $\dim_D(N)\leq2\dim_D(V)$ for any net $N\subseteq V$, where $N_\epsilon = \epsilon$-net means it is $\epsilon$-covering ($\forall v\in V, \exists n\in N_\epsilon$ s.t. $d(n,v)<\epsilon$) and $\epsilon$-separated (Take $n_1\neq n_2\in N_\epsilon$, $d(n_1,n_2)>\epsilon$)I would like to k...Read more

eventually and frequently for nets

I'm studying nets and there is something in the definition of eventually and frequently that is confusing me.These are the definitions I have.Eventually: a net $(x_\lambda)_{\lambda \in \Lambda}$ in a set $X$ is eventually in $Y \subset X$ if $\exists \tilde{\lambda} \in \Lambda$ such that $\forall \lambda \ge \tilde{\lambda}$, $x_\lambda \in Y$.Frequently: a net $(x_\lambda)_{\lambda \in \Lambda}$ in a set $X$ is frequently in $Y \subset X$ if $\forall \lambda \in \Lambda$, $\exists \mu \in \Lambda, \mu \ge \lambda$ such that $x_\mu \in Y$.Now...Read more