# Why Hahn-Banach theorem is needed for the following theorem?

One theorem in Rudin's Real and complex analysis says the following:

If $$X$$ is a normed linear space and if $$x_0 \in X$$, $$x_0\neq 0$$, there is a bounded linear funcitonal $$f$$ on $$X$$, of norm 1, so that $$f(x_0)=||x_0||$$.

He uses the Hahn-Banach theorem to prove it, but why we cannot just say that $$f(x)=||x||$$ works?

$$f(x)=\|x\|$$ is not linear!
There is no guarantee that $$f(x+y) = \|x+y\|$$ is equal to $$f(x)+f(y) = \|x\|+\|y\|$$.