In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof:
$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$ $\lVert x\rVert_\infty\le\lVert x\rVert_2\le\sqrt{n}\lVert x\rVert_\infty;$ $\lVert x\rVert_\infty\le\lVert x\rVert_1\le n\lVert x\rVert_\infty.$
Is that very easy to prove this inequality?
Also wanted to know when the equality is attained?