Proving the identity of a group and the identity of the subgroup are the same

Question

My question is suppose A is a subgroup of the group G. If e_G is the identity element of G and e_A is the identity element of A, how can I prove that e_G=e_A.

I was thinking of doing this by contradiction, but I am not too sure if this is the best approach.

Would a better choice be to use the identity property of groups?

Answer 1

Hint: See if you can prove that $e_Ge_A=e_Ae_A$. How can you then conclude that $e_G=e_A$?