In this work the rendezvous problem of multiagent systems with the event-triggered control scheme is studied. Each agent is assumed to have only a limited communication range. Connectivity-preserving rendezvous controllers with bounded inputs have been developed using multiple triggers and the constraint function approach. With some very mild assumptions, existing communication links can be preserved during the group rendezvous process. Moreover, some improvements have been introduced to avoid the Zeno behavior for each agent. The effectiveness of the proposed approach has been illustrated by simulations.