Using the triangular inequality (and the given hint!), we have
[tex]|x|=|(x-y)+y|\leq|x-y|+|y|\implies|x|-|y|\leq|x-y|[/tex]
Similarly,
[tex]|y|=|(y-x)+x|\leq|x-y|+|x|\implies|y|-|x|\leq|x-y|\implies -|x-y| \leq |x|-|y|[/tex]
We managed to bound the quantities in this fashion:
[tex]-b\leq a\leq b \implies |a|<b[/tex]
And thus we have
[tex]||x|-|y||\leq |x-y|[/tex]
