Chapter 3.2, Problem 27E

Suppose that $G$ is a nonempty set that is closed under an associative binary operation $\ast$ and that the following two conditions hold:

There exists a left identity $e$ in $G$ such that $e\ast x=x$ for all $x\in G$.

Each $a\in G$ has a left inverse $a_l$ in $G$ such that $a_l\ast a=e$.

Prove that $G$ is a group by showing that $e$ is in fact a two-sided identity for $G$ and that $a_l$ is a two-sided inverse of $a$.