Assumption of proof with contrapositive

Question

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 ?

Answer 1

The contrapositive of

If $∥u(0))∥=0$, then $‖u(t))‖=0$ for all $t$ belonging to $[0,T]$.

is the following:

If $‖u(t))‖\ne 0$ for some $t$ belonging to $[0,T]$, then $∥u(0))∥\ne 0$.

Answer 2

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 $\Vert u(0) \Vert \not= 0.$