I'm trying to prove that in a normed vector space $E$, a linear functional $L\colon E\to \mathbb{R}$ is continuous if $$\exists M\in \mathbb{R},\; \|L\|<M.$$
My attempt: Let $x\in E$ and $\varepsilon >0$. Let $x_0\in B(x,\varepsilon/M)$, then $|L(x)-L(x_0)|=|L(x-x_0)|$. Defining $y=x-x_0$, we have that $y\in B(0,\varepsilon/M)$. Then $\|y\|<\varepsilon /M$ and thus $\|y M/\varepsilon\|<1$ and since $\|L\|<M$, $|L(y M/\varepsilon)|<M$ and we conclude that $|L(y)|<\varepsilon$, that is $|L(x)-L(x_0)|<\varepsilon$ $\square$.
Is this proof correct?