A question about two definitions of covering numbers

Question

I am reading the book "Greedy approximation"(Page 208) written by V. Temlyakov and thinking about the following question of covering numbers:

Let $X$ be a Banach space and let $B_X(y,r)$ denote the ball of $X$ whose center is at $y$ with radius $r$.

Denote $N_{\epsilon}(A,X):=\min\{n: \exists y^1, y^2, ... ,y^n, where\ \ each \ \ y^j\in X: A\subset \cup_{j=1}^n B_X(y^j,\epsilon)\}$.

Denote $N(A,\epsilon,X):=\min\{n: \exists y^1, y^2, ... ,y^n, where\ \ each \ \ y^j\in A: A\subset \cup_{j=1}^n B_X(y^j,\epsilon)\}$.

Prove that: $$N_{\epsilon}(A,X)\leq N(A,\epsilon,X)\leq N_{{\epsilon}/2}(A,X)$$

I can understand that the first inequality but I am a bit of confusion about the second inequality.

My idea for the first inequality is because for each $j$, $y^j\in A\subset X$. This means the minimum cover number would be smaller if we consider a bigger set. I have no idea why the second inequality is correct.

Any suggestions would be welcome and thank you !!

Answer 1

• Suppose $y^1, \ldots, y^n \in X$ such that the $B_X(y^j, \epsilon/2)$ cover $A$.
• Each ball contains some point of $A$ (else you can throw away that ball to get a smaller cover). Let these points be $z^1, \ldots, z^n$.
• Use the triangle inequality to argue that the balls $B_X(z^j, \epsilon)$ cover $A$.