- #1

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## Homework Statement

Show that [tex]\|x\| \leq A|x| \forall x \in \mathbb{R},[/tex] where [tex]A \geq 0.[/tex]

## Homework Equations

We know the norm is a function [tex]f: {\mathbb{R}}^{d} \to \mathbb{R},[/tex] such that:

[tex]a) f(x) = 0 \iff x = 0,[/tex]

[tex]b) f(x+y) \leq f(x) + f(y),[/tex] and

[tex]c) f(cx) = |c|f(x) \forall c \in \mathbb{R}[/tex]

## The Attempt at a Solution

Ugh, I'm completely stumped here, and don't know where to begin. I know that for the Euclidean norm this is trivial, but I don't know how to even begin showing this in general. In particular, I don't see where I could grasp the absolute value (or the dot product), or how to start comparing it to the norm.

Any help would be greatly appreciated.