### 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