How did the founding of cities by Alexander ensure that Greek ideas and influence would continue on, even after his death?

Alexander the Great didn't just conquer vast swathes of territory; he wanted to leave behind a cultural legacy that would endure for centuries. And founding large cities was a very useful way of doing this. Most societies at that time were predominantly rural, but those cities that existed tended to be the prime transmitters of culture. So Alexander knew that if he could establish cities throughout his empire, he could set the tone of its cultural life long after his death.
As a proud inheritor of Hellenic culture, Alexander wanted to see that culture spread far and wide. And the way he did this was by establishing cities ruled by ethnic Greek elites. Using their enormous power, these rulers would enforce what they regarded as a superior culture on the indigenous people they'd conquered.
Probably the most famous example of this policy comes from Egypt, which was ruled by the Ptolemaic dynasty for almost three hundred years. The Ptolemies were, like Alexander himself, ethnic Greeks, and they brought Hellenic culture with them into their newly acquired territory. The Egyptian royal family spoke Greek rather than the native Egyptian language and actively disseminated Greek culture and learning, while at the same time they consolidated their rule by presenting themselves as heirs to the ancient pharaohs. It was this astute mixture of Greek and indigenous culture that ensured the Ptolemies' survival until the defeat of Cleopatra and Mark Antony at the Battle of Actium in 31 BCE.