If $\displaystyle a \leq b$ and $\displaystyle b \leq a+c$, then $\displaystyle |a-b| \leq c$.

Ok this is what I need to prove. I am fairly new to this so i'm not too sure what i have done is right, but here is what i have.

$\displaystyle a \leq b \leq a+c$ (subtract a)

$\displaystyle 0 \leq b-a \leq c$ (multiply by -1)

$\displaystyle 0 \geq a-b \geq -c$ so (??? not sure after this point ???) $\displaystyle c \geq 0$. therefor $\displaystyle c \geq 0 \geq a-b \geq -c$ so $\displaystyle c \geq a-b \geq -c$ which equals $\displaystyle |a-b| \leq c$.

Please if i have don something wrong correct me or if there is a better way of going about this that would be helpful too. Thanks for you help.