Originally Posted by
Jose27 Yes they're the same, and you actually have the definition that's useful in this case!
Remember that $\displaystyle \mathbb{Q} ^n$ is countable and ennumerate it by $\displaystyle (x_n)_{n\in \mathbb{N} }$. Given $\displaystyle a>0$ take cover the sphere (boundary of the ball) with sets of the form $\displaystyle \{ B_{r_{x_n}}(x_n) : r_{x_n} < \frac{a}{2^n} \}=A$ then $\displaystyle A$ certainly covers your sphere and since this last one is compact it can be covered by a finite number of these, say (after possibly renaming them) $\displaystyle B=(B_{r_{x_i}}(x_i))_{i=1}^k$ then $\displaystyle vol(sphere) \leq vol(\cup B ) \leq \sum_{i=1}^{\infty } \frac{a}{2^i} =a$ so the volume of the sphere is bounded by an arbitrarily small number.