Assumption of proof with contrapositive I want to prove by proof with contrapositive that
$\left \| u(0) \right \|=0 $ then $\left \| u(t) \right \|=0$ for all t belongs to $\left [ 0,T \right ]$.
Then I write
$\left \| u(t) \right \|\neq 0$ then $\left \| u(0) \right \|\neq 0$ for all t belongs to $\left [ 0,T \right ]$.
Is it correct ?

The contrapositive of "A implies B" is "not B implies not A". In this case A is $\Vert u(0) \Vert = 0$ and B is $\Vert u(t)   \Vert = 0$ for all $t \in [0, T].$ Not A is clearly $\Vert u(0) \Vert \not= 0.$ What about not B? If it is not true that something happens for every $t \in [0, T],$ it means that there is at least one value of $t$ in the interval for which that something doesn't happen. In other words, there exists $t_0 \in [0, T]$ such that $\Vert u(t_0) \Vert \not= 0.$ 
So what you have to prove is that if there exists $t_0 \in [0, T]$ such that $\Vert u(t_0) \Vert \not= 0,$ then $\Vert u(0) \Vert \not= 0.$
