Theorem: $\frac{H}{H \cap N} \simeq \frac{HN}{N}$. Define $\phi : H\rightarrow G/N$ by $\phi(h)=hN \ \ \ \forall h \in H$

I'm not sure why the image, $\phi(H) = HN/N$ Can anyone show me why? My book has

$$\phi(h) =\{gN \in G/N \ | \ gN=hN \ for \ some \ h\in H\}$$
$$ = \{ gN \in G/N \ | g \in HN\} = HN/N$$

But I don't see how and why, $gN= hN\rightarrow g \in HN$

Thanks :D!