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 in G such that .

definition 2: A group G is called residually finite if for all nontrivial element x in G, there is a homomorphism from G to a finite group such that .

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 be a residually finite group.

Let be a subgroup of and .

By def. 1, there exists a normal subgroup with finite index in such that .

Let be a normal subgroup with finite index in .

Since intersection of subgroups is also a subgroup, i have as a subgroup of .

So, now i have and .

I apply 2nd isomorphism theorem here and obtained and .

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

I want to prove that . Please give me some hints.

Thank you.