Suppose $\displaystyle N \unlhd G $ and $\displaystyle N \unlhd GN $.

I'm having trouble understanding why $\displaystyle GN/N \neq G/N $. Here's my reasoning:

Let $\displaystyle H = GN $, then $\displaystyle H/N = \{hN \mid h \in H \} = \{gnN \mid g \in G, n \in N \} = \{gN \mid g \in G \} = G/N $.

Where I used $\displaystyle nN = N $ since $\displaystyle n \in N $. What is wrong with this 'proof'?