Is it true that

$\Vert \left(\begin{array}{c}\hat{L}-L\\ G\end{array}\right)\Vert $

is minimized when $\hat{L}=L$ ($\hat{L}$ and $G$ are fixed, $L$ is the only variable quantity), i.e $\Vert \left(\begin{array}{c}\hat{L}-L\\ G\end{array}\right)\Vert =\Vert \left(\begin{array}{c}0\\ G\end{array}\right)\Vert $ ?

I'm unable to prove using just elementar inequalities such as triangle inequality or norm properties, any help would be appreciated.