Calculate

$\sqrt{2013^2 - 2(2012) + \dfrac{2012^2}{2013^2}} = \sqrt{\left(2013 - \dfrac{2012}{2013}\right) ^2}$

$\therefore$ question becomes $\left(2013 - \dfrac{2012}{2013} - \dfrac{1}{2013}\right) = 2012$

General computation: $\begin{aligned}\sqrt{2013-2(2012)+\frac{2012^2}{2013^2}}-\frac{1}{2013} &= \sqrt{4052169-4024+\frac{4048144}{4052169}}-\frac{1}{2013} \\&= \sqrt{\frac{1.640377\ce{e}+13}{4052169}}-\frac{1}{2013} \\&=\frac{\sqrt{1.640377\ce{e}+13}}{2013} -\frac{1}{2013} \\&= \frac{\sqrt{1.640377\ce{e}+13} -1}{2013} \\&=2012 \space \space \space \square \end{aligned}$

$\large \text{FIN!!!}$