Let $X$ be a Normed Linear Space and $x_{0} \in X, x_{0} \neq 0$. Show that there exists a bounded linear functional $\tilde{f}: X \rightarrow \mathbb{R}$ such that $$ ||\tilde{f}|| = 1 \, \, \, \text{and} \, \, \, \tilde{f}(x_{0}) = ||x_{0}|| $$
So far, here is what I have:
Let $X_{0} = span[x_{0}]$ and let $f: X_{0} \rightarrow \mathbb{R}$ be defined as $f(x) = f(\lambda x_{0}) = \lambda ||x_{0}||$.
It is clear that $f$ is bounded on $X_{0}$. Let $M = ||f|| = sup\lbrace |f(x)| : x \in X_{0}, \, ||x|| \leq 1 \rbrace$
We now define $p: X \rightarrow \mathbb{R} \, $ as $\, p(x) = M ||x||$. Note that $p$ is sub-additive, positively homogeneous and $f \leq p \, \text{ on } X_{0}$.
By Hahn-Banach, $f$ can be extended to a linear functional $\tilde{f}$ on all of $X$ and $$||\tilde{f}|| = 1, \, \, \tilde{f}(x_{0}) = ||x_{0}|| $$
I'm not sure what I'm missing, but it feels like I skipped some kind of important justification. Any assistance is appreciated!