$$ \begin{align} P{\lvert X-E(X) \rvert \geq\varepsilon} & =\int_{\lvert x-\mu \rvert \geq\varepsilon} f(x) , dx \ & \leq \int_{\lvert x-\mu \rvert \geq \varepsilon} \frac{\lvert x-\mu \rvert ^{2}}{\varepsilon^{2}}f(x) , dx \ & \leq \int_{-\infty}^{+\infty} \frac{\lvert x-\mu \rvert ^{2}}{\varepsilon^{2}}f(x) , dx \ & = \frac{1}{\varepsilon^{2}}\int_{-\infty}^{+\infty} \lvert x-\mu \rvert ^{2}f(x) , dx \ & =\frac{\sigma^{2}}{\varepsilon^{2}} \ & =\frac{D(X)}{\varepsilon^{2}} \end{align} $$