Infimum of elements in coset is the norm of that coset in quotient space

Question

I'm struggling with the following problem:

Let $Y$ be a closed subspace of a normed space of $(X,\|\cdot\|)$. Show that the norm $\|\cdot\|_0$ defined on $X/Y$ is given by $$\|\hat{x}\|_0 = \inf_{x\in\hat{x}}\|x\|$$ where $\hat{x} \in X/Y$.

I have previously shown that $\|\hat{x}\|_0 = p(x)$ for $x \in \hat{x}$ where $p$ is a seminorm on $X$. Using this, I think that $p(x) \leq \|x\| \, \forall x\in X$. So if I take a minimizing sequence $x_i \rightarrow \tilde{x} \in \hat{x}$ then $p(\tilde{x}) \leq \|\tilde{x}\|$ so that $\|\hat{x}\|_0 \leq \|\tilde{x}\|$.

1. I'm having a little trouble with the $\geq$ direction though.
2. I haven't actually shown $p(x) \leq \|x\| \, \forall x\in X$, I'm getting mixed up with this arguement. I'm pretty sure this is true though -- the seminorm takes non-zero vectors to $0$, the norm only takes the zero vector to $0$.