Norm of vectors inequality 
I tried proving this with triangular inequality but i was not right can any one help me with this 

We simply use the triangular inequality repeatedly:
$$
\|u\|=\|(u-v)+v\|\le \|u-v\|+\|v\|,
$$
and thus
$$
\|u\|-\|v\|\le \|u-v\|\le \|u\|+\|-v\|=\|u\|+\|v\|. \tag{1}
$$
Similarly
$$
\|v\|=\|(v-u)+u\|\le \|v-u\|+\|u\|,
$$
and thus
$$
\|v\|-\|u\|\le \|u-v\|. \tag{2}
$$
Now $(1)$ and $(2)$ imply that
$$
\big|\|u\|-\|v\|\big|\le \|u-v\|\le \|u\|+\|v\|.
$$
