---
tag: HM52W253FQ
kind: proof
name: Proof of the Ping-pong lemma for infinite cyclic subgroups
proves: J1PF2Q7KG1
sources:
  - wikipedia: en:18786361@r1361203357
    locator: Ping-pong lemma, section Formal statements > The Ping-pong lemma for infinite cyclic subgroups > Proof
    note: adopted
    mention: wikipedia://en/18786361#7PSC
    sections: [Proof]
---

## Proof

This statement follows as a corollary of the version for general subgroups if we let $X_{i} = X_{i}^{+} \cup X_{i}^{-}$ and let $H_{i} = \langle a_{i}\rangle$.