Originally Posted by

**ragnar** Define $\displaystyle ||\text{\bf x}|| = \sqrt{(x_{1})^{2} + ... + (x_{m})^{2}}$ for $\displaystyle \text{\bf x} \in \mathbb{R}^{m}$, and $\displaystyle ||\text{\bf x}||_{p} = \sqrt[p]{|x_{1}|^{p} + ... + |x_{m}|^{p}}$. Then prove what was said above: for any $\displaystyle m, p \geq 1$, there are scalars c, C > 0 such that for any $\displaystyle \text{\bf x} \in \mathbb{R}^{m}$ we have $\displaystyle c||\text{\bf x}|| \leq ||\text{\bf x}||_{p} \leq C||\text{\bf x}||$.