Understanding a proof exercise Evans PDE mean value formula

Question

I am trying to understand a solution to exercise 2.5 (3) in Evans' PDE pag 85-86, which tates the following:

The part in the Oval is the one I have trouble understanding. Thanks in advance for any help

Answer 1

Assuming $f$ is bounded, which is archived e.g. by $f\in C_c(B(0,r))$, say $|f|\leq K$.
For the volume of the n-dimensional ball with radius $\epsilon$, you have: $$ |B(0,\epsilon)|=\alpha(n)\epsilon^n $$ where $n$ is the dimension we are working in and $|\;|$ denotes the lebesgue measure.
So now you can estimate the integral: $$ \frac{1}{\epsilon^{n-2}}\int_{B(0, \epsilon)}fdy\leq \frac{1}{\epsilon^{n-2}} K |B(0,\epsilon)|=\alpha(n)K\frac{\epsilon^n}{\epsilon^{n-2}}=C\epsilon^2 $$