Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ {\mathbf P} \bigl\{ \max_{t\in[0,2]} \|W(t)\|> x\bigr\}\le 2 {\mathbf P} \bigl\{\chi_n^2>x^2/2\bigr\} $$ Why is this bound true? What other deviation inequalities for the Bessel processes are known?

Let $W(t)$ be an $n$-dimensional Wiener process. Denote by $\chi_n^2$ a chi-squared random variable with $n$ degrees of freedom. I have recently found the following inequality given without proof: $$ {\mathbf P} \left\{ \max_{t\in[0,2]} \|W(t)\|> x\right\}\leqslant 2 {\mathbf P} \bigl\{\chi_n^2>x^2/2\bigr\}. $$ Why is this bound true? What other deviation inequalities for the Bessel processes are known?