i need to show that subgroups of residually finite groups are residually finite.

i have 2 definitions for residually finite.

definition 1: A group G is called residually finite if for all nontrivial element in G, there exists a normal subgroup with finite index $\displaystyle N_x$ in G such that $\displaystyle x \notin N_x$.

definition 2: A group G is called residually finite if for all nontrivial element x in G, there is a homomorphism $\displaystyle f$ from G to a finite group such that $\displaystyle f(x)\ne1$.

NonCommAlg proved the above by using definition 2.

now i try to prove by using definition 1. im stuck somewhere and quite confused where to continue. please give me some hints.

here is my proof:

Let $\displaystyle G$ be a residually finite group.

Let $\displaystyle H$ be a subgroup of $\displaystyle G$ and $\displaystyle 1 \ne x \in H$.

By def. 1, there exists a normal subgroup with finite index $\displaystyle N$ in $\displaystyle G$ such that $\displaystyle x \notin N$.

Let $\displaystyle N'$ be a normal subgroup with finite index in $\displaystyle H$.

Since intersection of subgroups is also a subgroup, i have $\displaystyle N'$ as a subgroup of $\displaystyle G$.

So, now i have $\displaystyle N \vartriangleleft G$ and $\displaystyle N'<G$.

I apply 2nd isomorphism theorem here and obtained $\displaystyle N \cap N' ={\{1\}} \triangleleft N'$ and $\displaystyle N'/(N \cap N') \cong NN'/N$.

I don't know whether I followed the correct path, but i'm stuck here.

I want to prove that $\displaystyle x \notin N'$. Please give me some hints.

Thank you.