Let $X$ be a normed linear space and let $F$ be a linear functional defined on $X$. Prove that $F$ is unbounded if and only if for each $a \in X$ and each $r > 0,\, \{F(x): \|x-a\| < r \} = \mathbb{R}$.
I am particularly interested in one direction:
If for each $a \in X$ and each $r > 0,\,\{F(x): \|x-a\| < r \} = \mathbb{R}$ , then $F$ is unbounded.
Is the proof of this direction trivial because $\mathbb{R}$ is unbounded hence $F(x)$ is unbounded? Or is there a rigorous way of writing it?
We have (with $a=0$ and $r=1$), that $\{F(x): \|x\| < 1 \} = \mathbb{R}.$
Now suppose that $F$ is bounded. Then there is $c>0$ such that $|F(x)| \le c$ for all $x \in X$ with $||x|| \le 1.$ Hence
$$\{F(x): \|x\| < 1 \} \subseteq [-c,c],$$
a contradiction.