Every torsion-free group is a subgroup of a simple torsion-free group

This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.
View a complete list of embeddability theorems

Statement

Suppose $G$ is a Torsion-free group (?): in other words, no non-identity element of $G$ has finite order. Then, there exists a Simple torsion-free group (?) $L$ containing $G$.

Facts used

1. Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes

Proof

The proof follows directly from fact (1), and the observation that a group with two conjugacy classes is simple.