Proofs of Cauchy Inequality and Triangle Inequality for other norms on $\Bbb R^n$

Traditionally, for $\Bbb R^n$ with Euclidean inner prodct on it and $\ell^2$-norm on it, we first prove the Cauchy inequality $|x\cdot y|\leq|x|\cdot|y|$ for all $x,y\in\Bbb R^n$. Then we consequently prove the Triangle inequality $|x+y|\leq|x|+|y|$. All norms used here is the $\ell^2$-norm. However, I remember that $\Bbb R^n$ with $\ell^1$ or $\ell^\infty$ norms also form a normed vector space. Then how do we easily prove the one of the normed vector space axiom - Triangle inequality for these norms? Do they also satisfy the Cauchy inequality? Since the proof of the Cauchy inequality for the $\ell^2$ norm is very ad hoc to the $\ell^2$ norm, I can't see how to generalize it to other norms.

• What are the other classical inequalities in analysis besides Cauchy and Triangle? – Project Book Oct 21 '17 at 11:25
• It is proven directly. The theorem is called Minkowski inequality. – user251257 Oct 21 '17 at 12:41