Every norm in $\mathbb{R}$ has the form $||x||=a|x|$

Question

How can I prove the following?

Every norm in $\mathbb{R}$ has the form $||x||=a|x|$ where $a>0$ and $|x|$ is the absolute value of $x$. Conclude that every norm in $\mathbb{R}$ comes from a inner product.

First I think in prove by contradiction, but I don't know how to express that $||x||$ doesn't has the form $a|x|$, so maybe that's not the way.

I'm just starting to study metric spaces, and I really don't know how to solve this. If there is some hint to guide me I'll appreciate it.

Answer 1

Let $a=||1||>0$. So $||x||=||1\cdot x||=||1|| \,|x|=a|x|$.