An absolute inequality

Question

May I ask is the following correct?

I say, $\forall c,a,\delta\in\mathbb{R},|c-a|-|\delta|\leq |c+\delta-a|\leq |c-a|+|\delta|$ is correct.

Right side:

$\begin{align*} |c+\delta-a|&=|c-a+\delta|\\ &\leq |c-a|+|\delta|\text{ ,triangle inequality} \end{align*} $

Left side:

$\begin{align*} |c-a|-|\delta|&=|c-\delta+\delta-a|-|\delta|\\ &\leq |c+\delta-a|+|-\delta|-|\delta|\\ &= |c+\delta-a| \end{align*} $

Answer 1

Yes, all steps of your proof are correct.