Prove that in \(\mathbb{R}^{n}\) all \(\ell^{p}\)-norms \((1 \leqslant p \leqslant \infty)\) are equivalent:

\[\max _{1 \leqslant j \leqslant n}\left|x_{j}ight| \leqslant\left(\sum_{j=1}^{n}\left|x_{j}ight|^{p}ight)^{1 / p} \leqslant n^{1 / p} \max _{1 \leqslant j \leqslant n}\left|x_{j}ight| \quad \text { for all } x=\left(x_{1}, \ldots, x_{n}ight) \in \mathbb{R}^{n} .\]