Asking about proof of invertibility of matrix

Question

To prove: A matrix M is invertible if and only if zero is not an eigenvalue of M

proof: $\Leftarrow$: if zero is not an eigenvalue of $M$, then $v=0$ is the only vector such that $Mv=0$. This shows M is injective and thus $M$ is invertible.

$\Rightarrow$: If $M$ is invertible, then there exists no non-zero vector $u$ such that $M(u)=0 * u=0$

Question 1): From definition, eigenvector is never equal $0$. Why does this proof not violate the definition?

Question 2): From the $\Rightarrow$: I am confused, does here say that $u$ is a $0$ eigenvector? Why does it write $0 * u$ here?

Answer 1

There is no null eigenvector here.

For the $\implies$, note that if $0$ is an eigenvalue, then there is a nonzero eigenvector $v$ such that

$$Mv=0v=\vec0$$

But since $M$ is invertible,

$$v=M^{-1}Mv=M^{-1}\vec0=\vec0$$

Contradiction.