The phenomenon of high availability (HA) is of vital importance in cloud architecture. This paper proposes an HA verification framework, called HAVerifier, for OpenStack, a popular open source cloud platform. Fault injection technology has been adopted to verify the system's reliability by determining its health status after injecting faults. The framework proposed in this paper verifies the availability of services by injecting faults into the different components of OpenStack. Service indicators (for example, downtime) are monitored after the faults are injected. The collected metrics are compared with the provided service level agreement to verify whether the platform's availability meets the requirements. The fault injection steps can be implemented dynamically using the proposed framework, and the faults injected into the platform can be restored without manual intervention. Finally, a prototype for this framework is implemented to prove its applicability to verifying the HA of the OpenStack platform.