If we take the definition of a normal subgroup to be:

H is normal if $\displaystyle ghg^{-1} \in H $ for all h in H, g in G.

So it is clear that $\displaystyle gHg^{-1} \subset H $

However, I want to show that actually $\displaystyle gHg^{-1} = H $, by showing that $\displaystyle H \subset gHg^{-1} $

I'm just struggling to find a way to prove the final part? Could anyone offer any help?

Thanks,