I am trying to prove that if $X$ is complete and $ M \subset X$ has a finite epsilon net, then $M$ is relatively compact. I have the proof to hand:
I don't understand how they calculated that $\text{diam} T_j \leq 2\epsilon_j$. We have the $T_j$'s are all nested, then surely $T_1$ is the set with largest radius, and from the definition of $\text{diam} T_n = sup_{x_i,x_j \in T_n}d(x_i,x_j)$ surely we should have that $\text{diam} T_j \leq 2\epsilon_1$? any explanation please