Suppose that $\displaystyle A$ and $\displaystyle B$ are subgroups of a group $\displaystyle G$ with $\displaystyle A\subseteq B$ . Prove that $\displaystyle [G:B]$ divides $\displaystyle [G:A]$

Let N be a normal subgroup of a group G and let H be a subgroup of G that contains N

i) Show that $\displaystyle H/N$ is a subgroup of $\displaystyle G/N$

ii) If $\displaystyle H/N \triangleleft G/N$ prove that $\displaystyle H \triangleleft G$

I know these should be relatively straightforward and I have some answers but think they're really messy and not sure if what I've done is viable!

For first part using LaGrange I have $\displaystyle |G|=|G:B||B|$ and $\displaystyle |G| = |G:A||A|$ then as $\displaystyle A\subseteq B$ are both subgroups $\displaystyle |B| = |B:A||A|$ so $\displaystyle |G:B||B:A||A| = |G:A||A| \Rightarrow |G:B||B:A| = |G:A|$ so that $\displaystyle [G:B]$ divides $\displaystyle [G:A]$

For i) as all $\displaystyle h \in H$ also $\displaystyle \in G$ clearly$\displaystyle H/N \subseteq G/N$

to show it has an inverse say $\displaystyle h, h' \in H $ such that $\displaystyle hh'=e$ then $\displaystyle hNh'N = hh'NN = e$ (can do this as $\displaystyle N$ is normal)

then following same argument for $\displaystyle h, k \in H, hk \in H$ also so $\displaystyle hNkN=hkN \in H/N$

For ii) as $\displaystyle H/N \triangleleft G/N$ we have $\displaystyle gNhN(gN)^{-1}=gNhNg^{-1}N=(ghg^{-1})N \in H/N$ thus $\displaystyle ghg^{-1} \in H $ so $\displaystyle H \triangleleft G$

If anyone can let me know if what I've put is correct and or whether I've missed crucial steps I'd really appreciate it!

Thanks