I have 2 definitions for residually finite groups.

Definition 1: is said to be residually finite if for each nontrivial element, in , there exists a finite index normal subgroup in G such that .

Definition 2: is said to be residually finite if for each nontrivial element, , there exists a homomorphism from to a finite group, such thatf(x)is not the identity.

I have a problem in proving finite extension of residually finite groups are residually finite.

I have , with is residually finite and is finite.

Can anyone show me where to start or some hints?