Question

Prove that for all x in R^n and all multi-indices a,
|x^a| ≤ |x|^(|a|)

Answer 1

$|x^a|\leqslant|x|^{|a|} \ \ \forall x\isin\Reals^{n}, \ a=(a_1,a_2,...,a_n), a_n\geqslant0$

$|x^a|=|x_1^{a_1}x_2^{a_2}...x_n^{a_n}|=|x_1^{a_1}||x_2^{a_2}|...|x_n^{a_n}|=|x_1|^{a_1}|x_2|^{a_2}...|x_3|^{a_3}$

$|x|^{|a|}=|x|^{a_1+a_2+...+a_n}=|x|^{a_1}|x|^{a_2}...|x|^{a_n}=\sqrt{x_1^2+x_2^2+...+x_n^2}^{a_1}\sqrt{x_1^2+x_2^2+...+x_n^2}^{a_2}...\sqrt{x_1^2+x_2^2+...+x_n^2}^{a_n}$

$|x^a|\leqslant|x|^{|a|}$