I'm currently reading the paper [The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras](https://eudml.org/doc/153553) and having difficulty in understanding the proof of Proposition $4.5$ from the paper. 

> Let $A$ and $B$ be $C^{\ast}$-algebras and $I$ be a nonzero closed ideal of $A \otimes^{\text{min}} B$,  then $I$ contains a nonzero elementary tensor. 

Suppose not. Let $\pi: A \otimes^{\text{min}} B \to \frac{A \otimes^{\text{min}} B}{I}$ be the natural map, and define a $C^{\ast}$-seminorm as $N(u) = \| \pi(u) \|$ for $ u \in A \otimes^{\text{min}} B$. By assumption, $N(a\otimes b) > 0$ for all nonzero elementary tensors $a \otimes b$.

Then it's written that $N$ norm dominates the min norm, but I don't understand how $N$ is a norm. I can only see that it's a seminorm.


P.S: This question was first posted on [MSE](https://math.stackexchange.com/questions/4702288/difficulty-in-understanding-a-proof-from-an-article).