Prove that the identity element in a group is unique.
Suppose $e_1$ and $e_2$ are two identity elements of a group $G$, then for $g\in G$ we have $e_1g = ge_1 = e_1$ and $e_2g = ge_2 = e_2$ then $e_2e_1g = ge_1e_2 = e_2$. The proof in the book stops here, saying $e_1e_2 =e_2$.
This is the proof in the book, and I did the same in my attempt. However, the book follows the above by simply stating, "hence $e_1 = e_2$". How is this?