show that $ \lim \limits _{n\to\infty} \frac {Vol(\{ x \in B \mid \lvert x_1 \rvert > \epsilon \}} {Vol(B)} \to 0$ where $B$ is the unit ball

Question

given $\epsilon \gt 0$ show that:

$$\lim \limits _{n\to\infty} \frac{Vol(\{ x \in B \mid \lvert x_1 \rvert \gt \epsilon \}} {Vol(B)} \to 0$$

where $B$ is the unit ball in $n$ dimensions: $ B = \{ x \in \Bbb{R^n} \mid \lvert x \rvert \le 1 \}$.

Answer 1

Hint.

You have $$\mathrm{Vol}(B_n)= \frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}$$ and $$\mathrm{Vol}(\{ x \in B_n \mid \lvert x_1 \rvert > \epsilon\}) \le 2(1-\epsilon) \mathrm{Vol}(B_{n-1}) = 2 (1-\epsilon) \frac{\pi^{(n-1)/2}}{\Gamma(\frac{n-1}{2}+1)}$$