Let $f:X \rightarrow \mathbb R$ be a continuous linear functional. Prove $f$ is bounded.

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Since $f$ is continuous, $\forall \varepsilon >0$, there exists $\delta >0$ such that $|f(x)-f(y)|=|f(x-y)|=|f(z)|< \varepsilon$ whenever $|x-y|<\delta$. We let $z=x-y$.

Can we just now let $\varepsilon = C \|x\|\_X$ for some $C>0$ and then it is bounded?