Suppose that $x=(x_1,x_2,\ldots,x_n) \in \mathbb{R}^n$ and $w_1,w_2,\ldots,w_n>0$. Prove that $\|x\|_p=\left(\sum_{i=1}^n w_i |x_i|^p \right)^{\tfrac{1}{p}}$, $1 \le p < \infty$ is a norm in $\mathbb{R}^n$.
I tried to prove the triangle inequality which is $\|x\|_p + \|y\|_p \ge \|x+y\|_p$ but not succeed. Any help is appreciated. Thank you.
