"become satisfied with" is correct and usable in written English.
You can use it to describe someone achieving a feeling of contentment or level of satisfaction. For example, "After two years of hard work, Mary finally became satisfied with her career."
He sees his role as being someone whose job it is to push and prod and not let anyone become satisfied with the progress they have made.
I have to become satisfied with small successes and with completing the little tasks that can lead to big change, given time.
This is the proportion who could potentially become satisfied with the service as a result of each intervention.
For this range of T (see also Fig. 2) cooperation prevails, and hence individuals rapidly become satisfied with all their links.
But they don't always pursue her vision, either because they succeed at developing a single trait, and then, to her dismay, become satisfied with the result, or because they are out-and-out renegades.
They engage in role-playing activities, such as cooperative plays and pretending plays, and keep playing them until they become satisfied with their roles.
As Wielowiejski put it, this approach meant that many gay people became satisfied with being accepted as a minority and felt comfortable ignoring other social injustices that didn't directly affect them.
Being on time and staying until the job is complete becomes less important to us as we become less satisfied with the work we do.
When people inherit a large sum of money, for instance, they become more satisfied with their lives, according to recent research.
John Philpott, who carried out the study for the Chartered Institute of Personnel and Development, says: "Although older workers have become more satisfied with their pay and have a stronger sense of job security, they are a lot less happy with the intrinsic quality of their jobs and the hours they work".
